getfem_fem.cc

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/*===========================================================================

 Copyright (C) 1999-2017 Yves Renard

 This file is a part of GetFEM++

 GetFEM++  is  free software;  you  can  redistribute  it  and/or modify it
 under  the  terms  of the  GNU  Lesser General Public License as published
 by  the  Free Software Foundation;  either version 3 of the License,  or
 (at your option) any later version along with the GCC Runtime Library
 Exception either version 3.1 or (at your option) any later version.
 This program  is  distributed  in  the  hope  that it will be useful,  but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or  FITNESS  FOR  A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License and GCC Runtime Library Exception for more details.
 You  should  have received a copy of the GNU Lesser General Public License
 along  with  this program;  if not, write to the Free Software Foundation,
 Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA.

===========================================================================*/

/** @file getfem_fem.cc
    @author Yves Renard <Yves.Renard@insa-lyon.fr>
    @date December 21, 1999.
    @brief implementation of some finite elements.
 */

#include "getfem/bgeot_torus.h"
#include "getfem/dal_singleton.h"
#include "getfem/dal_tree_sorted.h"
#include "gmm/gmm_algobase.h"
#include "getfem/dal_naming_system.h"
#include "getfem/getfem_fem.h"
#include "getfem/getfem_integration.h"
#include "getfem/getfem_omp.h"
#include "getfem/getfem_torus.h"

namespace getfem {

  using bgeot::read_base_poly;
  using bgeot::base_rational_fraction;

  const base_matrix& fem_interpolation_context::M() const {
    if (gmm::mat_nrows(M_) == 0) {
      GMM_ASSERT2(have_pgt() && have_G() && have_pf(), "cannot compute M");
      M_.resize(pf_->nb_dof(convex_num()), pf_->nb_base(convex_num()));
      pf_->mat_trans(M_,G(),pgt());
    }
    return M_;
  }

  size_type fem_interpolation_context::convex_num() const {
    GMM_ASSERT3(convex_num_ != size_type(-1), "");
    return convex_num_;
  }

  bool fem_interpolation_context::is_convex_num_valid() const {
    return convex_num_ != size_type(-1);
  }

  short_type fem_interpolation_context::face_num() const {
    GMM_ASSERT3(face_num_ != short_type(-1),
                "Face number is asked but not defined");
    return face_num_;
  }

  bool fem_interpolation_context::is_on_face() const {
    return (face_num_ != short_type(-1));
  }

  // Comment regarding fem defined on the real element:
  // Precomputed values, gradients and hessians of fem defined on the real
  // element are not dealt with within fem_interpolation_context.
  // In that case, any precomputations can be performed within the fem
  // itself, which has to store the corresponding values internally.
  // The methods real_base_value, real_grad_base_value or real_hess_base_value
  // of the fem have the possibility to check if the passed ctx has a pfp
  // and extract the corresponding internally stored results based on
  // ctx.convex_num(), ctx.pfp()->get_ppoint_tab() and ctx.ii().
  // In that case, the storage available in ctx.pfp()->c, ctx.pfp()->pc
  // and ctx.pfp()->hpc is not used.


  // Specific multiplication for fem_interpolation_context use.
  static inline void spec_mat_tmult_(const base_tensor &g, const base_matrix &B,
                                     base_tensor &t) {
    size_type P = B.nrows(), N = B.ncols();
    size_type M = t.adjust_sizes_changing_last(g, P);
    bgeot::mat_tmult(&(*(g.begin())), &(*(B.begin())), &(*(t.begin())),M,N,P);
  }

  void fem_interpolation_context::pfp_base_value(base_tensor& t,
                                                 const pfem_precomp &pfp__) {
    const pfem &pf__ = pfp__->get_pfem();
    GMM_ASSERT1(ii_ != size_type(-1), "Internal error");

    if (pf__->is_standard())
      t = pfp__->val(ii());
    else {
      if (pf__->is_on_real_element())
        pf__->real_base_value(*this, t);
      else {
        switch(pf__->vectorial_type()) {
        case virtual_fem::VECTORIAL_NOTRANSFORM_TYPE:
          t = pfp__->val(ii()); break;
        case virtual_fem::VECTORIAL_PRIMAL_TYPE:
          t.mat_transp_reduction(pfp__->val(ii()), K(), 1); break;
        case virtual_fem::VECTORIAL_DUAL_TYPE:
          t.mat_transp_reduction(pfp__->val(ii()), B(), 1); break;
        }
        if (!(pf__->is_equivalent())) {
          set_pfp(pfp__);
          { base_tensor u = t; t.mat_transp_reduction(u, M(), 0); }
        }
      }
    }
  }


  void fem_interpolation_context::base_value(base_tensor& t,
                                             bool withM) const {
    if (pfp_ && ii_ != size_type(-1) && pf_->is_standard())
      t = pfp_->val(ii());
    else {
      if (pf_->is_on_real_element())
        pf_->real_base_value(*this, t);
      else {
        if (pfp_ && ii_ != size_type(-1)) {
          switch(pf_->vectorial_type()) {
          case virtual_fem::VECTORIAL_NOTRANSFORM_TYPE:
            t = pfp_->val(ii()); break;
          case virtual_fem::VECTORIAL_PRIMAL_TYPE:
            t.mat_transp_reduction(pfp_->val(ii()), K(), 1); break;
          case virtual_fem::VECTORIAL_DUAL_TYPE:
            t.mat_transp_reduction(pfp_->val(ii()), B(), 1); break;
          }
        }
        else {
          switch(pf_->vectorial_type()) {
          case virtual_fem::VECTORIAL_NOTRANSFORM_TYPE:
            pf_->base_value(xref(), t); break;
          case virtual_fem::VECTORIAL_PRIMAL_TYPE:
            {
              base_tensor u; pf_->base_value(xref(), u);
              t.mat_transp_reduction(u,K(),1);
            } break;
          case virtual_fem::VECTORIAL_DUAL_TYPE:
            {
              base_tensor u; pf_->base_value(xref(), u);
              t.mat_transp_reduction(u,B(),1);
            } break;
          }
        }
        if (withM && !(pf_->is_equivalent()))
          { base_tensor u = t; t.mat_transp_reduction(u, M(), 0); }
      }
    }
  }

  void fem_interpolation_context::pfp_grad_base_value
  (base_tensor& t, const pfem_precomp &pfp__) {
    const pfem &pf__ = pfp__->get_pfem();
    GMM_ASSERT1(ii_ != size_type(-1), "Internal error");

    if (pf__->is_standard()) {
      // t.mat_transp_reduction(pfp__->grad(ii()), B(), 2);
      spec_mat_tmult_(pfp__->grad(ii()), B(), t);
    } else {
      if (pf__->is_on_real_element())
        pf__->real_grad_base_value(*this, t);
      else {
        switch(pf__->vectorial_type()) {
        case virtual_fem::VECTORIAL_PRIMAL_TYPE:
          {
            base_tensor u;
            // u.mat_transp_reduction(pfp__->grad(ii()), B(), 2);
            spec_mat_tmult_(pfp__->grad(ii()), B(), u);
            t.mat_transp_reduction(u, K(), 1);
          }
          break;
        case virtual_fem::VECTORIAL_DUAL_TYPE:
          {
            base_tensor u;
            // u.mat_transp_reduction(pfp__->grad(ii()), B(), 2);
            spec_mat_tmult_(pfp__->grad(ii()), B(), u);
            t.mat_transp_reduction(u, B(), 1);
          }
          break;
        default:
          // t.mat_transp_reduction(pfp__->grad(ii()), B(), 2);
          spec_mat_tmult_(pfp__->grad(ii()), B(), t);
        }
        if (!(pf__->is_equivalent())) {
          set_pfp(pfp__);
          base_tensor u = t; t.mat_transp_reduction(u, M(), 0);
        }
      }
    }
  }


  void fem_interpolation_context::grad_base_value(base_tensor& t,
                                                  bool withM) const {
    if (pfp_ && ii_ != size_type(-1) && pf_->is_standard()) {
      // t.mat_transp_reduction(pfp_->grad(ii()), B(), 2);
      spec_mat_tmult_(pfp_->grad(ii()), B(), t);
    } else {
      if (pf()->is_on_real_element())
        pf()->real_grad_base_value(*this, t);
      else {
        if (have_pfp() && ii() != size_type(-1)) {
          switch(pf()->vectorial_type()) {
          case virtual_fem::VECTORIAL_PRIMAL_TYPE:
            {
              base_tensor u;
              // u.mat_transp_reduction(pfp_->grad(ii()), B(), 2);
              spec_mat_tmult_(pfp_->grad(ii()), B(), u);
              t.mat_transp_reduction(u, K(), 1);
            }
            break;
          case virtual_fem::VECTORIAL_DUAL_TYPE:
            {
              base_tensor u;
              // u.mat_transp_reduction(pfp_->grad(ii()), B(), 2);
              spec_mat_tmult_(pfp_->grad(ii()), B(), u);
              t.mat_transp_reduction(u, B(), 1);
            }
            break;
          default:
            // t.mat_transp_reduction(pfp_->grad(ii()), B(), 2);
            spec_mat_tmult_(pfp_->grad(ii()), B(), t);
          }

        } else {
          base_tensor u;
          pf()->grad_base_value(xref(), u);
          if (u.size()) { /* only if the FEM can provide grad_base_value */
            // t.mat_transp_reduction(u, B(), 2);
            spec_mat_tmult_(u, B(), t);
            switch(pf()->vectorial_type()) {
            case virtual_fem::VECTORIAL_PRIMAL_TYPE:
              u = t; t.mat_transp_reduction(u, K(), 1); break;
            case virtual_fem::VECTORIAL_DUAL_TYPE:
              u = t; t.mat_transp_reduction(u, B(), 1);  break;
            default: break;
            }
          }
        }
        if (withM && !(pf()->is_equivalent()))
          { base_tensor u = t; t.mat_transp_reduction(u, M(), 0); }
      }
    }
  }

  void fem_interpolation_context::hess_base_value(base_tensor& t,
                                                  bool withM) const {
    if (pf()->is_on_real_element())
      pf()->real_hess_base_value(*this, t);
    else {
      base_tensor tt;
      if (have_pfp() && ii() != size_type(-1))
        tt = pfp()->hess(ii());
      else
        pf()->hess_base_value(xref(), tt);

      switch(pf()->vectorial_type()) {
      case virtual_fem::VECTORIAL_PRIMAL_TYPE:
        { base_tensor u = tt; tt.mat_transp_reduction(u, K(), 1); } break;
      case virtual_fem::VECTORIAL_DUAL_TYPE:
        { base_tensor u = tt; tt.mat_transp_reduction(u, B(), 1); } break;
      default: break;
      }

      if (tt.size()) { /* only if the FEM can provide hess_base_value */
        tt.adjust_sizes(tt.sizes()[0], tt.sizes()[1], gmm::sqr(tt.sizes()[2]));
        t.mat_transp_reduction(tt, B3(), 2);
        if (!pgt()->is_linear()) {
          if (have_pfp()) {
            tt.mat_transp_reduction(pfp()->grad(ii()), B32(), 2);
          } else {
            base_tensor u;
            pf()->grad_base_value(xref(), u);
            tt.mat_transp_reduction(u, B32(), 2);
          }
          t -= tt;
        }
        if (!(pf()->is_equivalent()) && withM)
          { tt = t; t.mat_transp_reduction(tt, M(), 0); }
      }
    }
  }

  void fem_interpolation_context::set_pfp(pfem_precomp newpfp) {
    if (pfp_ != newpfp) {
      pfp_ = newpfp;
      if (pfp_) { pf_ = pfp()->get_pfem(); }
      else pf_ = 0;
      M_.resize(0,0);
    }
  }

  void fem_interpolation_context::set_pf(pfem newpf) {
    if (pf_ != newpf || have_pfp()) {
      set_pfp(0);
      pf_ = newpf;
    }
  }

  void virtual_fem::real_base_value(const fem_interpolation_context &c,
                                    base_tensor &t, bool withM) const
  { c.base_value(t, withM); }

  void virtual_fem::real_grad_base_value(const fem_interpolation_context &c,
                                    base_tensor &t, bool withM) const
  { c.grad_base_value(t, withM);}

  void virtual_fem::real_hess_base_value(const fem_interpolation_context &c,
                                         base_tensor &t, bool withM) const
  { c.hess_base_value(t, withM); }

  /* ******************************************************************** */
  /*        Class for description of an interpolation dof.                */
  /* ******************************************************************** */

  enum ddl_type { LAGRANGE, NORMAL_DERIVATIVE, DERIVATIVE, MEAN_VALUE,
                  BUBBLE1, LAGRANGE_NONCONFORMING, GLOBAL_DOF,
                  SECOND_DERIVATIVE, NORMAL_COMPONENT, EDGE_COMPONENT};

  struct ddl_elem {
    ddl_type t;
    gmm::int16_type hier_degree;
    short_type hier_raff;
    bool operator < (const ddl_elem &l) const {
      if (t < l.t) return true;
      if (t > l.t) return false;
      if (hier_degree < l.hier_degree) return true;
      if (hier_degree > l.hier_degree) return false;
      if (hier_raff < l.hier_raff) return true;
      return false;
    }
    ddl_elem(ddl_type s = LAGRANGE, gmm::int16_type k = -1, short_type l = 0)
      : t(s), hier_degree(k), hier_raff(l) {}
  };

  struct dof_description {
    std::vector<ddl_elem> ddl_desc;
    bool linkable;
    dim_type coord_index;
    size_type xfem_index;
    bool all_faces;

    dof_description()
    { linkable = true; all_faces = false; coord_index = 0; xfem_index = 0; }
  };

  struct dof_description_comp__ {
    int operator()(const dof_description &m, const dof_description &n) const;
  };

  // ATTENTION : en cas de modif, changer aussi dof_description_compare,
  //             product_dof, et dof_hierarchical_compatibility.
  int dof_description_comp__::operator()(const dof_description &m,
                                         const dof_description &n) const {
    int nn = gmm::lexicographical_less<std::vector<ddl_elem> >()
      (m.ddl_desc, n.ddl_desc);
    if (nn < 0) return -1;
    if (nn > 0) return 1;
    nn = int(m.linkable) - int(n.linkable);
    if (nn < 0) return -1;
    if (nn > 0) return 1;
    nn = int(m.coord_index) - int(n.coord_index);
    if (nn < 0) return -1;
    if (nn > 0) return 1;
    nn = int(m.xfem_index) - int(n.xfem_index);
    if (nn < 0) return -1;
    if (nn > 0) return 1;
    nn = int(m.all_faces) - int(n.all_faces);
    if (nn < 0) return -1;
    if (nn > 0) return 1;
    return 0;
  }

  typedef dal::dynamic_tree_sorted<dof_description, dof_description_comp__> dof_d_tab;

  pdof_description lagrange_dof(dim_type n) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(dim_type, n_old, dim_type(-2));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pdof_description, p_old, 0);
    if (n != n_old) {
      dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
      dof_description l;
      l.ddl_desc.resize(n);
      std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(LAGRANGE));
      p_old = &(tab[tab.add_norepeat(l)]);
      n_old = n;
    }
    return p_old;
  }

  pdof_description lagrange_0_dof(dim_type n) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(dim_type, n_old, dim_type(-2));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pdof_description, p_old, 0);
    if (n != n_old) {
      dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
      dof_description l;
      l.all_faces = true;
      l.ddl_desc.resize(n);
      l.linkable = false;
      std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(LAGRANGE));
      p_old = &(tab[tab.add_norepeat(l)]);
      n_old = n;
    }
    return p_old;
  }

  pdof_description deg_hierarchical_dof(pdof_description p, int deg) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l = *p;
    for (size_type i = 0; i < l.ddl_desc.size(); ++i)
      l.ddl_desc[i].hier_degree = gmm::int16_type(deg);
    return &(tab[tab.add_norepeat(l)]);
  }

  size_type reserve_xfem_index() {
    static size_type ind = 100;
    return ind += 1000;
  }

  pdof_description xfem_dof(pdof_description p, size_type ind) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l = *p; l.xfem_index = ind;
    return &(tab[tab.add_norepeat(l)]);
  }


  pdof_description to_coord_dof(pdof_description p, dim_type ct) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l = *p;
    l.coord_index = ct;
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description raff_hierarchical_dof(pdof_description p, short_type deg) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l = *p;
    for (size_type i = 0; i < l.ddl_desc.size(); ++i)
      l.ddl_desc[i].hier_raff = deg;
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description lagrange_nonconforming_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l; l.linkable = false;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(LAGRANGE));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description bubble1_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(BUBBLE1));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description derivative_dof(dim_type n, dim_type num_der) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(LAGRANGE));
    l.ddl_desc.at(num_der) = ddl_elem(DERIVATIVE);
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description second_derivative_dof(dim_type n, dim_type num_der1,
                                         dim_type num_der2) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(LAGRANGE));
    l.ddl_desc[num_der1] = ddl_elem(SECOND_DERIVATIVE);
    l.ddl_desc[num_der2] = ddl_elem(SECOND_DERIVATIVE);
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description normal_derivative_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(),
              ddl_elem(NORMAL_DERIVATIVE));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description normal_component_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(),
              ddl_elem(NORMAL_COMPONENT));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description edge_component_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(),
              ddl_elem(EDGE_COMPONENT));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description mean_value_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.ddl_desc.resize(n);
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(), ddl_elem(MEAN_VALUE));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description global_dof(dim_type n) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    dof_description l;
    l.all_faces = true;
    l.ddl_desc.resize(n);
    l.linkable = false;
    std::fill(l.ddl_desc.begin(), l.ddl_desc.end(),ddl_elem(GLOBAL_DOF));
    return &(tab[tab.add_norepeat(l)]);
  }

  pdof_description product_dof(pdof_description a, pdof_description b) {
    dof_d_tab& tab = dal::singleton<dof_d_tab>::instance();
    size_type nb1 = a->ddl_desc.size(), nb2 = b->ddl_desc.size();
    dof_description l;
    l.linkable = a->linkable && b->linkable;
    l.coord_index = std::max(a->coord_index, b->coord_index); // logique ?
    l.xfem_index = a->xfem_index;
    l.all_faces = a->all_faces || b->all_faces;
    GMM_ASSERT2(a->xfem_index == b->xfem_index, "Invalid product of dof");
    l.ddl_desc.resize(nb1+nb2);
    std::copy(a->ddl_desc.begin(), a->ddl_desc.end(), l.ddl_desc.begin());
    std::copy(b->ddl_desc.begin(), b->ddl_desc.end(), l.ddl_desc.begin()+nb1);

    {
      gmm::int16_type deg = -1;
      for (size_type i = 0; i < l.ddl_desc.size(); ++i)
        deg = std::max(deg, l.ddl_desc[i].hier_degree);
      for (size_type i = 0; i < l.ddl_desc.size(); ++i)
        l.ddl_desc[i].hier_degree = deg;
    }
    {
      short_type deg = 0;
      for (size_type i = 0; i < l.ddl_desc.size(); ++i)
        deg = std::max(deg, l.ddl_desc[i].hier_raff);
      for (size_type i = 0; i < l.ddl_desc.size(); ++i)
        l.ddl_desc[i].hier_raff = deg;
    }
    return &(tab[tab.add_norepeat(l)]);
  }

  // ATTENTION : en cas de modif, changer aussi
  //             dof_description_comp__::operator,
  //             product_dof, et dof_hierarchical_compatibility.
  //             et dof_description_compare
  int dof_weak_compatibility(pdof_description a, pdof_description b) {
    int nn;
    std::vector<ddl_elem>::const_iterator
      ita = a->ddl_desc.begin(), itae = a->ddl_desc.end(),
      itb = b->ddl_desc.begin(), itbe = b->ddl_desc.end();
    for (; ita != itae && itb != itbe; ++ita, ++itb)
    {
      if ((nn = int(ita->t) - int (itb->t)) != 0) return nn;
      if ((nn = int(ita->hier_degree) - int (itb->hier_degree)) != 0)
        return nn;
    }
    for (; ita != itae; ++ita) if (ita->t != LAGRANGE) return 1;
    for (; itb != itbe; ++itb) if (itb->t != LAGRANGE) return -1;
    return 0;
  }


  // ATTENTION : en cas de modif, changer aussi
  //             dof_description_comp__::operator,
  //             product_dof, et dof_hierarchical_compatibility.
  int dof_description_compare(pdof_description a, pdof_description b) {
    if (a == b) return 0;
    int nn;
    if ((nn = int(a->coord_index) - int(b->coord_index)) != 0) return nn;
    if ((nn = int(a->linkable) - int(b->linkable)) != 0) return nn;
    if ((nn = int(a->xfem_index) - int(b->xfem_index)) != 0) return nn;
    return dof_weak_compatibility(a,b);
  }

  bool dof_linkable(pdof_description a)
  { return a->linkable; }

  bool dof_compatibility(pdof_description a, pdof_description b)
  { return (dof_linkable(a) && dof_description_compare(a, b) == 0); }

  size_type dof_xfem_index(pdof_description a)
  { return a->xfem_index; }

  dim_type coord_index_of_dof(pdof_description a)
  { return a->coord_index; }

  bool dof_hierarchical_compatibility(pdof_description a, pdof_description b)
  {
    if (a->coord_index != b->coord_index) return false;
    if (a->linkable != b->linkable) return false;
    if (a->xfem_index != b->xfem_index) return false;
    std::vector<ddl_elem>::const_iterator
      ita = a->ddl_desc.begin(), itae = a->ddl_desc.end(),
      itb = b->ddl_desc.begin(), itbe = b->ddl_desc.end();
    for (; ita != itae && itb != itbe; ++ita, ++itb)
    { if (ita->t != itb->t) return false; }
    for (; ita != itae; ++ita) if (ita->t != LAGRANGE) return false;
    for (; itb != itbe; ++itb) if (itb->t != LAGRANGE) return false;
    return true;
  }


  /* ******************************************************************** */
  /*        Members methods of virtual_fem           .                    */
  /* ******************************************************************** */


  void virtual_fem::add_node(const pdof_description &d, const base_node &pt,
                             const dal::bit_vector &faces) {
    short_type nb = cv_node.nb_points();
    cv_node.points().resize(nb+1);
    cv_node.points()[nb] = pt;
    dof_types_.resize(nb+1);
    face_tab.resize(nb+1);
    dof_types_[nb] = d;
    cvs_node->add_point_adaptative(nb, short_type(-1));
    for (dal::bv_visitor f(faces); !f.finished(); ++f) {
      cvs_node->add_point_adaptative(nb, short_type(f));
      face_tab[nb].push_back(short_type(f));
    }
    pspt_valid = false;
  }


  void virtual_fem::add_node(const pdof_description &d, const base_node &pt) {
    dal::bit_vector faces;
    for (short_type f = 0; f < cvs_node->nb_faces(); ++f)
      if (d->all_faces || gmm::abs(cvr->is_in_face(f, pt)) < 1.0E-7)
        faces.add(f);
     add_node(d, pt, faces);
  }

  void virtual_fem::init_cvs_node() {
    cvs_node->init_for_adaptative(cvr->structure());
    cv_node = bgeot::convex<base_node>(cvs_node);
    face_tab.resize(0);
    pspt_valid = false;
  }

  void virtual_fem::unfreeze_cvs_node() {
    cv_node.structure() = cvs_node;
    pspt_valid = false;
  }

  const std::vector<short_type> &
  virtual_fem::faces_of_dof(size_type /*cv*/, size_type i) const {
    static const std::vector<short_type> no_faces;
    return (i < face_tab.size()) ? face_tab[i] : no_faces;
  }

  void virtual_fem::copy(const virtual_fem &f) {
    dof_types_ = f.dof_types_;
    cvs_node = bgeot::new_convex_structure();
    *cvs_node = *f.cvs_node; // deep copy
    cv_node = f.cv_node;
    cv_node.structure() = cvs_node;
    pspt = 0;
    pspt_valid = false;
    cvr = f.cvr;
    dim_ = f.dim_;
    ntarget_dim = f.ntarget_dim;
    vtype = f.vtype;
    is_equiv = f.is_equiv;
    is_lag = f.is_lag;
    is_pol = f.is_pol;
    is_polycomp = f.is_polycomp;
    real_element_defined = f.real_element_defined;
    is_standard_fem = f.is_standard_fem;
    es_degree = f.es_degree;
    hier_raff = f.hier_raff;
    debug_name_ = f.debug_name_;
    face_tab = f.face_tab;
  }

  /* ******************************************************************** */
  /*        PK class.                                                         */
  /* ******************************************************************** */

  class PK_fem_ : public fem<base_poly> {
  public :
    void calc_base_func(base_poly &p, size_type i, short_type K) const;
    PK_fem_(dim_type nc, short_type k);
    ~PK_fem_() {}
  };

  void PK_fem_::calc_base_func(base_poly &p, size_type i, short_type K) const {
    dim_type N = dim();
    base_poly l0(N, 0), l1(N, 0);
    bgeot::power_index w(short_type(N+1));
    l0.one(); l1.one(); p = l0;

    if (K != 0) {
      for (short_type nn = 0; nn < N; ++nn) l0 -= base_poly(N, 1, nn);

      w[0] = K;
      for (short_type nn = 1; nn <= N; ++nn) {
        w[nn]=short_type(floor(0.5+bgeot::to_scalar((cv_node.points()[i])[nn-1]*opt_long_scalar_type(K))));
        w[0]=short_type(w[0] - w[nn]);
      }

      for (int nn = 0; nn <= N; ++nn)
        for (int j = 0; j < w[nn]; ++j) {
          if (nn == 0)
            p *= (l0 * (opt_long_scalar_type(K) / opt_long_scalar_type(j+1)))
              - (l1 * (opt_long_scalar_type(j) / opt_long_scalar_type(j+1)));
          else
            p *= (base_poly(N,1,short_type(nn-1)) * (opt_long_scalar_type(K)/opt_long_scalar_type(j+1)))
              - (l1 * (opt_long_scalar_type(j) / opt_long_scalar_type(j+1)));
        }
    }
  }

  PK_fem_::PK_fem_(dim_type nc, short_type k) {
    cvr = bgeot::simplex_of_reference(nc);
    dim_ = cvr->structure()->dim();
    is_standard_fem = is_equiv = is_pol = is_lag = true;
    es_degree = k;

    init_cvs_node();
    bgeot::pconvex_ref cvn = bgeot::simplex_of_reference(nc, k);
    size_type R = cvn->nb_points();
    for (size_type i = 0; i < R; ++i)
      add_node(k==0 ? lagrange_0_dof(nc) : lagrange_dof(nc), cvn->points()[i]);

    base_.resize(R);
    for (size_type r = 0; r < R; r++) calc_base_func(base_[r], r, k);
  }

  static pfem PK_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = dim_type(::floor(params[0].num() + 0.01));
    int k = short_type(::floor(params[1].num() + 0.01));
    GMM_ASSERT1(n > 0 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    pfem p = std::make_shared<PK_fem_>(dim_type(n), short_type(k));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        Tensorial product of fem (for polynomial fem).                    */
  /* ******************************************************************** */

  struct tproduct_femi : public fem<base_poly> {
    tproduct_femi(ppolyfem fi1, ppolyfem fi2);
  };

  tproduct_femi::tproduct_femi(ppolyfem fi1, ppolyfem fi2) {
    if (fi2->target_dim() != 1) std::swap(fi1, fi2);
    GMM_ASSERT1(fi2->target_dim() == 1, "dimensions mismatch");

    is_pol = true;
    is_equiv = fi1->is_equivalent() && fi2->is_equivalent();
    GMM_ASSERT1(is_equiv,
                "Product of non equivalent elements not available, sorry.");
    is_lag = fi1->is_lagrange() && fi2->is_lagrange();
    is_standard_fem = fi1->is_standard() && fi2->is_standard();
    es_degree = short_type(fi1->estimated_degree() + fi2->estimated_degree());

    bgeot::convex<base_node> cv
      = bgeot::convex_direct_product(fi1->node_convex(0), fi2->node_convex(0));
    cvr = bgeot::convex_ref_product(fi1->ref_convex(0), fi2->ref_convex(0));
    dim_ = cvr->structure()->dim();
    init_cvs_node();

    ntarget_dim = fi2->target_dim();
    base_.resize(cv.nb_points() * ntarget_dim);
    size_type i, j, r;
    for (j = 0, r = 0; j < fi2->nb_dof(0); ++j)
      for (i = 0; i < fi1->nb_dof(0); ++i, ++r)
        add_node(product_dof(fi1->dof_types()[i], fi2->dof_types()[j]),
                 cv.points()[r]);

    for (j = 0, r = 0; j < fi2->nb_base_components(0); j++)
      for (i = 0; i < fi1->nb_base_components(0); i++, ++r) {
        base_[r] = fi1->base()[i];
        base_[r].direct_product(fi2->base()[j]);
      }
  }

  static pfem product_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 1 && params[1].type() == 1,
                "Bad type of parameters");
    pfem pf1 = params[0].method();
    pfem pf2 = params[1].method();
    GMM_ASSERT1(pf1->is_polynomial() && pf2->is_polynomial(),
                "Both arguments to FEM_PRODUCT must be polynomial FEM");
    pfem p = std::make_shared<tproduct_femi>(ppolyfem(pf1.get()),
                                             ppolyfem(pf2.get()));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*    Generic Hierarchical fem (for polynomial fem). To be interfaced.  */
  /* ******************************************************************** */

  struct thierach_femi : public fem<base_poly> {
    thierach_femi(ppolyfem fi1, ppolyfem fi2);
  };

  thierach_femi::thierach_femi(ppolyfem fi1, ppolyfem fi2)
    : fem<base_poly>(*fi1) {
    grad_computed_ = false;
    hess_computed_ = false;
    GMM_ASSERT1(fi2->target_dim()==fi1->target_dim(), "dimensions mismatch.");
    GMM_ASSERT1(fi2->basic_structure(0) == fi1->basic_structure(0),
                "Incompatible elements.");
    GMM_ASSERT1(fi1->is_equivalent() &&  fi2->is_equivalent(), "Sorry, "
                "no hierachical construction for non tau-equivalent fems.");
    es_degree = fi2->estimated_degree();
    is_lag = false;
    unfreeze_cvs_node();
    for (size_type i = 0; i < fi2->nb_dof(0); ++i) {
      bool found = false;
      for (size_type j = 0; j < fi1->nb_dof(0); ++j) {
        if ( gmm::vect_dist2(fi2->node_of_dof(0,i),
                             fi1->node_of_dof(0,j)) < 1e-13
             && dof_hierarchical_compatibility(fi2->dof_types()[i],
                                               fi1->dof_types()[j]))
            { found = true; break; }
      }
      if (!found) {
        add_node(deg_hierarchical_dof(fi2->dof_types()[i],
                                      fi1->estimated_degree()),
                 fi2->node_of_dof(0,i));
        base_.resize(nb_dof(0));
        base_[nb_dof(0)-1] = (fi2->base())[i];
      }
    }
  }

  struct thierach_femi_comp : public fem<bgeot::polynomial_composite> {
    thierach_femi_comp(ppolycompfem fi1, ppolycompfem fi2);
  };

  thierach_femi_comp::thierach_femi_comp(ppolycompfem fi1, ppolycompfem fi2)
    : fem<bgeot::polynomial_composite>(*fi1) {
    GMM_ASSERT1(fi2->target_dim()==fi1->target_dim(), "dimensions mismatch.");
    GMM_ASSERT1(fi2->basic_structure(0) == fi1->basic_structure(0),
                "Incompatible elements.");
    GMM_ASSERT1(fi1->is_equivalent() &&  fi2->is_equivalent(), "Sorry, "
                "no hierachical construction for non tau-equivalent fems.");

    es_degree = std::max(fi2->estimated_degree(), fi1->estimated_degree());

    is_lag = false;
    hier_raff = short_type(fi1->hierarchical_raff() + 1);
    unfreeze_cvs_node();
    for (size_type i = 0; i < fi2->nb_dof(0); ++i) {
      bool found = false;
      for (size_type j = 0; j < fi1->nb_dof(0); ++j) {
        if ( gmm::vect_dist2(fi2->node_of_dof(0,i),
                             fi1->node_of_dof(0,j)) < 1e-13
             && dof_hierarchical_compatibility(fi2->dof_types()[i],
                                               fi1->dof_types()[j]))
          { found = true; break; }
      }
      if (!found) {
        add_node(raff_hierarchical_dof(fi2->dof_types()[i], hier_raff),
                 fi2->node_of_dof(0,i));
        base_.resize(nb_dof(0));
        base_[nb_dof(0)-1] = (fi2->base())[i];
      }
    }
  }

  static pfem gen_hierarchical_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 1 && params[1].type() == 1,
                "Bad type of parameters");
    pfem pf1 = params[0].method();
    pfem pf2 = params[1].method();
    if (pf1->is_polynomial() && pf2->is_polynomial())
      return std::make_shared<thierach_femi>(ppolyfem(pf1.get()),
                                             ppolyfem(pf2.get()));
    GMM_ASSERT1(pf1->is_polynomialcomp() && pf2->is_polynomialcomp(),
                "Bad parameters");
    pfem p  = std::make_shared<thierach_femi_comp>(ppolycompfem(pf1.get()),
                                                   ppolycompfem(pf2.get()));
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /* PK hierarchical fem.                                                 */
  /* ******************************************************************** */

  static pfem PK_hierarch_fem(fem_param_list &params,
                              std::vector<dal::pstatic_stored_object> &) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01)), s;
    GMM_ASSERT1(n > 0 && n < 100 && k > 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (k == 1)
      name << "FEM_PK(" << n << ",1)";
    else {
      for (s = 2; s <= k; ++s) if ((k % s) == 0) break;
      name << "FEM_GEN_HIERARCHICAL(FEM_PK_HIERARCHICAL(" << n << ","
           << k/s << "), FEM_PK(" << n << "," << k << "))";
    }
    return fem_descriptor(name.str());
  }

  static pfem QK_hierarch_fem(fem_param_list &params,
                              std::vector<dal::pstatic_stored_object> &) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    GMM_ASSERT1(n > 0 && n < 100 && k > 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (n == 1)
      name << "FEM_PK_HIERARCHICAL(1," << k << ")";
    else
      name << "FEM_PRODUCT(FEM_PK_HIERARCHICAL(" << n-1 << "," << k
           << "),FEM_PK_HIERARCHICAL(1," << k << "))";
    return fem_descriptor(name.str());
  }

  static pfem prism_PK_hierarch_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (n == 2)
      name << "FEM_QK_HIERARCHICAL(1," << k << ")";
    else
      name << "FEM_PRODUCT(FEM_PK_HIERARCHICAL(" << n-1 << "," << k
           << "),FEM_PK_HIERARCHICAL(1," << k << "))";
    return fem_descriptor(name.str());
  }


  /* ******************************************************************** */
  /* parallelepiped fems.                                                 */
  /* ******************************************************************** */

  static pfem QK_fem_(fem_param_list &params, bool discontinuous) {
    const char *fempk = discontinuous ? "FEM_PK_DISCONTINUOUS" : "FEM_PK";
    const char *femqk = discontinuous ? "FEM_QK_DISCONTINUOUS" : "FEM_QK";
    GMM_ASSERT1(params.size() == 2 || (discontinuous && params.size() == 3),
                "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0 &&
                (params.size() != 3 || params[2].type() == 0),
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    char alpha[128]; alpha[0] = 0;
    if (discontinuous && params.size() == 3) {
      scalar_type v = params[2].num();
      GMM_ASSERT1(v >= 0 && v <= 1, "Bad value for alpha: " << v);
      sprintf(alpha, ",%g", v);
    }
    GMM_ASSERT1(n > 0 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (n == 1)
      name << fempk << "(1," << k << alpha << ")";
    else
      name << "FEM_PRODUCT(" << femqk << "(" << n-1 << ","
           << k << alpha << ")," << fempk << "(1," << k << alpha << "))";
    return fem_descriptor(name.str());
  }

  static pfem QK_fem(fem_param_list &params,
                     std::vector<dal::pstatic_stored_object> &) {
    return QK_fem_(params, false);
  }

  static pfem QK_discontinuous_fem(fem_param_list &params,
                                   std::vector<dal::pstatic_stored_object> &) {
    return QK_fem_(params, true);
  }


  /* ******************************************************************** */
  /* prims fems.                                                          */
  /* ******************************************************************** */

  static pfem prism_PK_fem(fem_param_list &params,
                           std::vector<dal::pstatic_stored_object> &) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (n == 2)
      name << "FEM_QK(1," << k << ")";
    else
      name << "FEM_PRODUCT(FEM_PK(" << n-1 << "," << k << "),FEM_PK(1,"
           << k << "))";
    return fem_descriptor(name.str());
  }

  static pfem
  prism_PK_discontinuous_fem(fem_param_list &params,
                             std::vector<dal::pstatic_stored_object> &) {
    GMM_ASSERT1(params.size() == 2 || params.size() == 3,
                "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0 &&
                (params.size() != 3 || params[2].type() == 0),
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    char alpha[128]; alpha[0] = 0;
    if (params.size() == 3) {
      scalar_type v = params[2].num();
      GMM_ASSERT1(v >= 0 && v <= 1, "Bad value for alpha: " << v);
      sprintf(alpha, ",%g", v);
    }
    GMM_ASSERT1(n > 1 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    std::stringstream name;
    if (n == 2)
      name << "FEM_QK_DISCONTINUOUS(1," << k << alpha << ")";
    else
      name << "FEM_PRODUCT(FEM_PK_DISCONTINUOUS(" << n-1 << "," << k << alpha
           << "),FEM_PK_DISCONTINUOUS(1,"
           << k << alpha << "))";
    return fem_descriptor(name.str());
  }

  /* ******************************************************************** */
  /*        P1 NON CONFORMING (dim 2)                                         */
  /* ******************************************************************** */

   static pfem P1_nonconforming_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters ");
    auto p = std::make_shared<fem<base_poly>>();
    p->mref_convex() = bgeot::simplex_of_reference(2);
    p->dim() = 2;
    p->is_standard() = p->is_equivalent() = true;
    p->is_polynomial() = p->is_lagrange() = true;
    p->estimated_degree() = 1;
    p->init_cvs_node();
    p->base().resize(3);

    p->add_node(lagrange_dof(2), base_small_vector(0.5, 0.5));
    read_poly(p->base()[0], 2, "2*x + 2*y - 1");
    p->add_node(lagrange_dof(2), base_small_vector(0.0, 0.5));
    read_poly(p->base()[1], 2, "1 - 2*x");
    p->add_node(lagrange_dof(2), base_small_vector(0.5, 0.0));
    read_poly(p->base()[2], 2, "1 - 2*y");
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));

    return pfem(p);
  }


  /* ******************************************************************** */
  /*     Quad8/Hexa20 SERENDIPITY ELEMENT (dim 2 or 3) (incomplete Q2)    */
  /* ******************************************************************** */

  // local dof numeration for 2D:
  // 5--6--7
  // |     |
  // 3     4
  // |     |
  // 0--1--2
  //
  // local dof numeration for 3D:
  //
  //       17---18---19
  //      /|        /|
  //     / 10      / 11
  //   15  |      16 |
  //   /   5----6/---7
  //  /   /     /   /
  // 12---13---14  /
  // |  3      |  4
  // 8 /       9 /
  // |/        |/
  // 0----1----2

   static pfem
   build_Q2_incomplete_fem(fem_param_list &params,
                           std::vector<dal::pstatic_stored_object> &deps,
                           bool discontinuous) {
    GMM_ASSERT1(params.size() <= 1, "Bad number of parameters");
    dim_type n = 2;
    if (params.size() > 0) {
      GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
      n = dim_type(::floor(params[0].num() + 0.01));
       GMM_ASSERT1(n == 2 || n == 3, "Bad parameter, expected value 2 or 3");
    }
    auto p = std::make_shared<fem<base_poly>>();
    p->mref_convex() = bgeot::parallelepiped_of_reference(n);
    p->dim() = n;
    p->is_standard() = p->is_equivalent() = true;
    p->is_polynomial() = p->is_lagrange() = true;
    p->estimated_degree() = 2;
    p->init_cvs_node();
    p->base().resize(n == 2 ? 8 : 20);
    auto lag_dof = discontinuous ? lagrange_nonconforming_dof(n) 
                                 : lagrange_dof(n);

    if (n == 2) {
      std::stringstream s
        ( "1 - 2*x^2*y - 2*x*y^2 + 2*x^2 + 5*x*y + 2*y^2 - 3*x - 3*y;"
          "4*(x^2*y - x^2 - x*y + x);"
          "2*x*y*y - 2*x*x*y + 2*x*x - x*y - x;"
          "4*(x*y*y - x*y - y*y + y);"
          "4*(x*y - x*y*y);"
          "2*x*x*y - 2*x*y*y - x*y + 2*y*y - y;"
          "4*(x*y - x*x*y);"
          "2*x*x*y + 2*x*y*y - 3*x*y;");

      for (int i = 0; i < 8; ++i) p->base()[i] = read_base_poly(2, s);

      p->add_node(lag_dof, base_small_vector(0.0, 0.0));
      p->add_node(lag_dof, base_small_vector(0.5, 0.0));
      p->add_node(lag_dof, base_small_vector(1.0, 0.0));
      p->add_node(lag_dof, base_small_vector(0.0, 0.5));
      p->add_node(lag_dof, base_small_vector(1.0, 0.5));
      p->add_node(lag_dof, base_small_vector(0.0, 1.0));
      p->add_node(lag_dof, base_small_vector(0.5, 1.0));
      p->add_node(lag_dof, base_small_vector(1.0, 1.0));
    } else {
      std::stringstream s
        ("1 + 2*x^2*y*z + 2*x*y^2*z + 2*x*y*z^2"
           " - 2*x^2*y - 2*x^2*z - 2*x*y^2 - 2*y^2*z - 2*y*z^2 - 2*x*z^2 - 7*x*y*z"
           " + 2*x^2 + 2*y^2 + 2*z^2 + 5*y*z + 5*x*z + 5*x*y - 3*x - 3*y - 3*z;"
         "4*( - x^2*y*z + x*y*z + x^2*z - x*z + x^2*y - x*y - x^2 + x);"
         "2*x^2*y*z - 2*x*y^2*z - 2*x*y*z^2"
           " - 2*x^2*y - 2*x^2*z + 2*x*y^2 + 2*x*z^2 + 3*x*y*z + 2*x^2 - x*y - x*z - x;"
         "4*( - x*y^2*z + x*y^2 + y^2*z + x*y*z - x*y - y^2 - y*z + y);"
         "4*(x*y^2*z - x*y^2 - x*y*z + x*y);"
         " - 2*x^2*y*z + 2*x*y^2*z - 2*x*y*z^2"
           " + 2*x^2*y - 2*x*y^2 - 2*y^2*z + 2*y*z^2 + 3*x*y*z - x*y + 2*y^2 - y*z - y;"
         "4*(x^2*y*z - x^2*y - x*y*z + x*y);"
         " - 2*x^2*y*z - 2*x*y^2*z + 2*x*y*z^2 + 2*x^2*y + 2*x*y^2 + x*y*z - 3*x*y;"
         "4*( - x*y*z^2 + x*z^2 + y*z^2 + x*y*z - x*z - y*z - z^2 + z);"
         "4*(x*y*z^2 - x*y*z - x*z^2 + x*z);"
         "4*(x*y*z^2 - x*y*z - y*z^2 + y*z);"
         "4*( - x*y*z^2 + x*y*z);"
         " - 2*x^2*y*z - 2*x*y^2*z + 2*x*y*z^2"
           " + 2*x^2*z + 2*y^2*z - 2*x*z^2 - 2*y*z^2 + 3*x*y*z - x*z - y*z + 2*z^2 - z;"
         "4*(x^2*y*z - x^2*z - x*y*z + x*z);"
         " - 2*x^2*y*z + 2*x*y^2*z - 2*x*y*z^2 + 2*x^2*z + 2*x*z^2 + x*y*z - 3*x*z;"
         "4*(x*y^2*z - y^2*z - x*y*z + y*z);"
         "4*( - x*y^2*z + x*y*z);"
         "2*x^2*y*z - 2*x*y^2*z - 2*x*y*z^2 + 2*y^2*z + 2*y*z^2 + x*y*z - 3*y*z;"
         "4*( - x^2*y*z + x*y*z);"
         "2*x^2*y*z + 2*x*y^2*z + 2*x*y*z^2 - 5*x*y*z;");

      for (int i = 0; i < 20; ++i) p->base()[i] = read_base_poly(3, s);

      p->add_node(lag_dof, base_small_vector(0.0, 0.0, 0.0));
      p->add_node(lag_dof, base_small_vector(0.5, 0.0, 0.0));
      p->add_node(lag_dof, base_small_vector(1.0, 0.0, 0.0));
      p->add_node(lag_dof, base_small_vector(0.0, 0.5, 0.0));
      p->add_node(lag_dof, base_small_vector(1.0, 0.5, 0.0));
      p->add_node(lag_dof, base_small_vector(0.0, 1.0, 0.0));
      p->add_node(lag_dof, base_small_vector(0.5, 1.0, 0.0));
      p->add_node(lag_dof, base_small_vector(1.0, 1.0, 0.0));

      p->add_node(lag_dof, base_small_vector(0.0, 0.0, 0.5));
      p->add_node(lag_dof, base_small_vector(1.0, 0.0, 0.5));
      p->add_node(lag_dof, base_small_vector(0.0, 1.0, 0.5));
      p->add_node(lag_dof, base_small_vector(1.0, 1.0, 0.5));

      p->add_node(lag_dof, base_small_vector(0.0, 0.0, 1.0));
      p->add_node(lag_dof, base_small_vector(0.5, 0.0, 1.0));
      p->add_node(lag_dof, base_small_vector(1.0, 0.0, 1.0));
      p->add_node(lag_dof, base_small_vector(0.0, 0.5, 1.0));
      p->add_node(lag_dof, base_small_vector(1.0, 0.5, 1.0));
      p->add_node(lag_dof, base_small_vector(0.0, 1.0, 1.0));
      p->add_node(lag_dof, base_small_vector(0.5, 1.0, 1.0));
      p->add_node(lag_dof, base_small_vector(1.0, 1.0, 1.0));
    }
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));

    return pfem(p);
  }

  static pfem Q2_incomplete_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps)
  { return build_Q2_incomplete_fem(params, deps, false); }

  static pfem Q2_incomplete_discontinuous_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps)
  { return build_Q2_incomplete_fem(params, deps, true); }

  /* ******************************************************************** */
  /*    Lagrange Pyramidal element of degree 0, 1 and 2                   */
  /* ******************************************************************** */

  // local dof numeration for K=1:
  //    4
  //   /|||
  //  / || |
  // 2-|--|-3
  // | |  | |
  // ||    ||
  // ||    ||
  // 0------1
  //
  // local dof numeration for K=2:
  //
  //    13
  //   /  |
  //  11---12
  //  |    |
  //  9----10
  //  /     |
  // 6---7---8
  // |       |
  // 3   4   5
  // |       |
  // 0---1---2

  static pfem
  build_pyramid_QK_fem(short_type k, bool disc, scalar_type alpha=0) {
    auto p = std::make_shared<fem<base_rational_fraction>>();
    p->mref_convex() = bgeot::pyramid_QK_of_reference(1);
    p->dim() = 3;
    p->is_standard() = p->is_equivalent() = true;
    p->is_polynomial() = false;
    p->is_lagrange() = true;
    p->estimated_degree() = k;
    p->init_cvs_node();
    auto lag_dof = disc ? lagrange_nonconforming_dof(3) : lagrange_dof(3);

    if (k == 0) {
      p->base().resize(1);
      p->base()[0] = read_base_poly(3, "1");
      p->add_node(lagrange_0_dof(3), base_small_vector(0.0, 0.0, 0.5));
    } else if (k == 1) {
      p->base().resize(5);
      base_rational_fraction // Q = xy/(1-z)
        Q(read_base_poly(3, "x*y"), read_base_poly(3, "1-z"));
      p->base()[0] = (read_base_poly(3, "1-x-y-z") + Q)*0.25;
      p->base()[1] = (read_base_poly(3, "1+x-y-z") - Q)*0.25;
      p->base()[2] = (read_base_poly(3, "1-x+y-z") - Q)*0.25;
      p->base()[3] = (read_base_poly(3, "1+x+y-z") + Q)*0.25;
      p->base()[4] = read_base_poly(3, "z");

      p->add_node(lag_dof, base_small_vector(-1.0, -1.0, 0.0));
      p->add_node(lag_dof, base_small_vector( 1.0, -1.0, 0.0));
      p->add_node(lag_dof, base_small_vector(-1.0,  1.0, 0.0));
      p->add_node(lag_dof, base_small_vector( 1.0,  1.0, 0.0));
      p->add_node(lag_dof, base_small_vector( 0.0,  0.0, 1.0));

    } else if (k == 2) {
      p->base().resize(14);

      base_poly xi0  = read_base_poly(3, "(1-z-x)*0.5");
      base_poly xi1  = read_base_poly(3, "(1-z-y)*0.5");
      base_poly xi2  = read_base_poly(3, "(1-z+x)*0.5");
      base_poly xi3  = read_base_poly(3, "(1-z+y)*0.5");
      base_poly x    = read_base_poly(3, "x");
      base_poly y    = read_base_poly(3, "y");
      base_poly z    = read_base_poly(3, "z");
      base_poly ones = read_base_poly(3, "1");
      base_poly un_z = read_base_poly(3, "1-z");

      std::vector<base_node> points = { base_node(-1.0, -1.0, 0.0),
                                        base_node( 0.0, -1.0, 0.0),
                                        base_node( 1.0, -1.0, 0.0),
                                        base_node(-1.0,  0.0, 0.0),
                                        base_node( 0.0,  0.0, 0.0),
                                        base_node( 1.0,  0.0, 0.0),
                                        base_node(-1.0,  1.0, 0.0),
                                        base_node( 0.0,  1.0, 0.0),
                                        base_node( 1.0,  1.0, 0.0),
                                        base_node(-0.5, -0.5, 0.5),
                                        base_node( 0.5, -0.5, 0.5),
                                        base_node(-0.5,  0.5, 0.5),
                                        base_node( 0.5,  0.5, 0.5),
                                        base_node( 0.0,  0.0, 1.0) };

      if (disc && alpha != scalar_type(0)) {
        base_node G =
          gmm::mean_value(points.begin(), points.end());
        for (auto && pt : points)
          pt = (1-alpha)*pt + alpha*G;
        for (size_type d = 0; d < 3; ++d) {
          base_poly S(1,2);
          S[0] = -alpha * G[d] / (1-alpha);
          S[1] = 1. / (1-alpha);
          xi0 = bgeot::poly_substitute_var(xi0, S, d);
          xi1 = bgeot::poly_substitute_var(xi1, S, d);
          xi2 = bgeot::poly_substitute_var(xi2, S, d);
          xi3 = bgeot::poly_substitute_var(xi3, S, d);
          x = bgeot::poly_substitute_var(x, S, d);
          y = bgeot::poly_substitute_var(y, S, d);
          z = bgeot::poly_substitute_var(z, S, d);
          un_z = bgeot::poly_substitute_var(un_z, S, d);
        }
      }
      base_rational_fraction Q(read_base_poly(3, "1"), un_z);

      p->base()[ 0] = Q*Q*xi0*xi1*(x*y-z*un_z);
      p->base()[ 1] = -Q*Q*xi0*xi1*xi2*y*4.;
      p->base()[ 2] = Q*Q*xi1*xi2*(-x*y-z*un_z);
      p->base()[ 3] = -Q*Q*xi3*xi0*xi1*x*4.;
      p->base()[ 4] = Q*Q*xi0*xi1*xi2*xi3*16.;
      p->base()[ 5] = Q*Q*xi1*xi2*xi3*x*4.;
      p->base()[ 6] = Q*Q*xi3*xi0*(-x*y-z*un_z);
      p->base()[ 7] = Q*Q*xi2*xi3*xi0*y*4.;
      p->base()[ 8] = Q*Q*xi2*xi3*(x*y-z*un_z);
      p->base()[ 9] = Q*z*xi0*xi1*4.;
      p->base()[10] = Q*z*xi1*xi2*4.;
      p->base()[11] = Q*z*xi3*xi0*4.;
      p->base()[12] = Q*z*xi2*xi3*4.;
      p->base()[13] = z*(z*2-ones);

      for (const auto &pt : points)
        p->add_node(lag_dof, pt);

    } else GMM_ASSERT1(false, "Sorry, pyramidal Lagrange fem "
                       "implemented only for degree 0, 1 or 2");

    return pfem(p);
  }


  static pfem pyramid_QK_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() <= 1, "Bad number of parameters");
    short_type k = 2;
    if (params.size() > 0) {
      GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
      k = dim_type(::floor(params[0].num() + 0.01));
    }
    pfem p = build_pyramid_QK_fem(k, false);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  static pfem pyramid_QK_disc_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() <= 2, "Bad number of parameters");
    short_type k = 2;
    if (params.size() > 0) {
      GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
      k = dim_type(::floor(params[0].num() + 0.01));
    }
    scalar_type alpha(0);
    if (params.size() > 1) {
      GMM_ASSERT1(params[1].type() == 0, "Bad type of parameters");
      alpha = params[1].num();
    }
    pfem p = build_pyramid_QK_fem(k, true, alpha);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*    Incomplete quadratic Lagrange pyramidal element.                  */
  /* ******************************************************************** */

  // local dof numeration:
  //
  //    12
  //   /  |
  //  10---11
  //  |    |
  //  8----9
  //  /     |
  // 5---6---7
  // |       |
  // 3       4
  // |       |
  // 0---1---2

  static pfem build_pyramid_Q2_incomplete_fem(bool disc) {
    auto p = std::make_shared<fem<base_rational_fraction>>();
    p->mref_convex() = bgeot::pyramid_QK_of_reference(1);
    p->dim() = 3;
    p->is_standard() = p->is_equivalent() = true;
    p->is_polynomial() = false;
    p->is_lagrange() = true;
    p->estimated_degree() = 2;
    p->init_cvs_node();
    auto lag_dof = disc ? lagrange_nonconforming_dof(3) : lagrange_dof(3);

    p->base().resize(13);

    base_poly xy = read_base_poly(3, "x*y");
    base_poly z = read_base_poly(3, "z");

    base_poly p00m = read_base_poly(3, "1-z");

    base_poly pmmm = read_base_poly(3, "1-x-y-z");
    base_poly ppmm = read_base_poly(3, "1+x-y-z");
    base_poly pmpm = read_base_poly(3, "1-x+y-z");
    base_poly pppm = read_base_poly(3, "1+x+y-z");

    base_poly mmm0_ = read_base_poly(3, "-1-x-y")*0.25;
    base_poly mpm0_ = read_base_poly(3, "-1+x-y")*0.25;
    base_poly mmp0_ = read_base_poly(3, "-1-x+y")*0.25;
    base_poly mpp0_ = read_base_poly(3, "-1+x+y")*0.25;

    base_poly pp0m = read_base_poly(3, "1+x-z");
    base_poly pm0m = read_base_poly(3, "1-x-z");
    base_poly p0mm = read_base_poly(3, "1-y-z");
    base_poly p0pm = read_base_poly(3, "1+y-z");

    // (Bedrosian, 1992)
    p->base()[ 0] = mmm0_ * pmmm + base_rational_fraction(mmm0_ * xy, p00m); // N5
    p->base()[ 1] = base_rational_fraction(pp0m * pm0m * p0mm * 0.5, p00m);  // N6
    p->base()[ 2] = mpm0_ * ppmm - base_rational_fraction(mpm0_ * xy, p00m); // N7
    p->base()[ 3] = base_rational_fraction(p0pm * p0mm * pm0m * 0.5, p00m);  // N4
    p->base()[ 4] = base_rational_fraction(p0pm * p0mm * pp0m * 0.5, p00m);  // N8
    p->base()[ 5] = mmp0_ * pmpm - base_rational_fraction(mmp0_ * xy, p00m); // N3
    p->base()[ 6] = base_rational_fraction(pp0m * pm0m * p0pm * 0.5, p00m);  // N2
    p->base()[ 7] = mpp0_ * pppm + base_rational_fraction(mpp0_ * xy, p00m); // N1
    p->base()[ 8] = base_rational_fraction(pm0m * p0mm * z, p00m);           // N11
    p->base()[ 9] = base_rational_fraction(pp0m * p0mm * z, p00m);           // N12
    p->base()[10] = base_rational_fraction(pm0m * p0pm * z, p00m);           // N10
    p->base()[11] = base_rational_fraction(pp0m * p0pm * z, p00m);           // N9
    p->base()[12] = read_base_poly(3, "2*z^2-z");                            // N13

    p->add_node(lag_dof, base_small_vector(-1.0, -1.0, 0.0));
    p->add_node(lag_dof, base_small_vector( 0.0, -1.0, 0.0));
    p->add_node(lag_dof, base_small_vector( 1.0, -1.0, 0.0));
    p->add_node(lag_dof, base_small_vector(-1.0,  0.0, 0.0));
    p->add_node(lag_dof, base_small_vector( 1.0,  0.0, 0.0));
    p->add_node(lag_dof, base_small_vector(-1.0,  1.0, 0.0));
    p->add_node(lag_dof, base_small_vector( 0.0,  1.0, 0.0));
    p->add_node(lag_dof, base_small_vector( 1.0,  1.0, 0.0));
    p->add_node(lag_dof, base_small_vector(-0.5, -0.5, 0.5));
    p->add_node(lag_dof, base_small_vector( 0.5, -0.5, 0.5));
    p->add_node(lag_dof, base_small_vector(-0.5,  0.5, 0.5));
    p->add_node(lag_dof, base_small_vector( 0.5,  0.5, 0.5));
    p->add_node(lag_dof, base_small_vector( 0.0,  0.0, 1.0));

    return pfem(p);
  }


  static pfem pyramid_Q2_incomplete_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = build_pyramid_Q2_incomplete_fem(false);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  static pfem pyramid_Q2_incomplete_disc_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = build_pyramid_Q2_incomplete_fem(true);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*    Incomplete quadratic Lagrange prism element.                      */
  /* ******************************************************************** */

  // local dof numeration:
  //
  //    14
  //    /|`
  //  12 | 13
  //  /  8  `
  // 9--10--11
  // |   |   |
  // |   5   |
  // 6  / `  7
  // | 3   4 |
  // |/     `|
  // 0---1---2

  static pfem build_prism_incomplete_P2_fem(bool disc) {
    auto p = std::make_shared<fem<base_rational_fraction>>();
    p->mref_convex() = bgeot::prism_of_reference(3);
    p->dim() = 3;
    p->is_standard() = p->is_equivalent() = true;
    p->is_polynomial() = false;
    p->is_lagrange() = true;
    p->estimated_degree() = 2;
    p->init_cvs_node();
    auto lag_dof = disc ? lagrange_nonconforming_dof(3) : lagrange_dof(3);

    p->base().resize(15);

    std::stringstream s
      ( "-2*y*z^2-2*x*z^2+2*z^2-2*y^2*z-4*x*y*z+5*y*z-2*x^2*z+5*x*z"
          "-3*z+2*y^2+4*x*y-3*y+2*x^2-3*x+1;"
        "4*(x*y*z+x^2*z-x*z-x*y-x^2+x);"
        "2*x*z^2-2*x^2*z-x*z+2*x^2-x;"
        "4*(y^2*z+x*y*z-y*z-y^2-x*y+y);"
        "4*(x*y-x*y*z);"
        "2*y*z^2-2*y^2*z-y*z+2*y^2-y;"
        "4*(y*z^2+x*z^2-z^2-y*z-x*z+z);"
        "4*(x*z-x*z^2);"
        "4*(y*z-y*z^2);"
        "-2*y*z^2-2*x*z^2+2*z^2+2*y^2*z+4*x*y*z-y*z+2*x^2*z-x*z-z;"
        "4*(-x*y*z-x^2*z+x*z);"
        "2*x*z^2+2*x^2*z-3*x*z;"
        "4*(-y^2*z-x*y*z+y*z);"
        "4*x*y*z;"
        "2*y*z^2+2*y^2*z-3*y*z;");

    for (int i = 0; i < 15; ++i)
      p->base()[i] = read_base_poly(3, s);

    p->add_node(lag_dof, base_small_vector(0.0, 0.0, 0.0));
    p->add_node(lag_dof, base_small_vector(0.5, 0.0, 0.0));
    p->add_node(lag_dof, base_small_vector(1.0, 0.0, 0.0));
    p->add_node(lag_dof, base_small_vector(0.0, 0.5, 0.0));
    p->add_node(lag_dof, base_small_vector(0.5, 0.5, 0.0));
    p->add_node(lag_dof, base_small_vector(0.0, 1.0, 0.0));
    p->add_node(lag_dof, base_small_vector(0.0, 0.0, 0.5));
    p->add_node(lag_dof, base_small_vector(1.0, 0.0, 0.5));
    p->add_node(lag_dof, base_small_vector(0.0, 1.0, 0.5));
    p->add_node(lag_dof, base_small_vector(0.0, 0.0, 1.0));
    p->add_node(lag_dof, base_small_vector(0.5, 0.0, 1.0));
    p->add_node(lag_dof, base_small_vector(1.0, 0.0, 1.0));
    p->add_node(lag_dof, base_small_vector(0.0, 0.5, 1.0));
    p->add_node(lag_dof, base_small_vector(0.5, 0.5, 1.0));
    p->add_node(lag_dof, base_small_vector(0.0, 1.0, 1.0));

    return pfem(p);
  }


  static pfem prism_incomplete_P2_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = build_prism_incomplete_P2_fem(false);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  static pfem prism_incomplete_P2_disc_fem
  (fem_param_list &params, std::vector<dal::pstatic_stored_object> &deps) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = build_prism_incomplete_P2_fem(true);
    deps.push_back(p->ref_convex(0));
    deps.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*        P1 element with a bubble base fonction on a face              */
  /* ******************************************************************** */

  struct P1_wabbfoaf_ : public PK_fem_ {
    P1_wabbfoaf_(dim_type nc);
  };

  P1_wabbfoaf_::P1_wabbfoaf_(dim_type nc) : PK_fem_(nc, 1) {
    is_lag = false; es_degree = 2;
    base_node pt(nc); pt.fill(0.5);
    unfreeze_cvs_node();
    add_node(bubble1_dof(nc), pt);
    base_.resize(nb_dof(0));
    base_[nc+1] = base_[1]; base_[nc+1] *= scalar_type(1 << nc);
    for (int i = 2; i <= nc; ++i) base_[nc+1] *= base_[i];
    // Le raccord assure la continuite
    // des possibilités de raccord avec du P2 existent mais il faudrait
    // modifier qlq chose (transformer les fct de base P1)
  }

  static pfem P1_with_bubble_on_a_face(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && double(n) == params[0].num(),
                "Bad parameter");
    pfem p  = std::make_shared<P1_wabbfoaf_>(dim_type(n));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*        Element RT0 on the simplexes.                                     */
  /* ******************************************************************** */

  struct P1_RT0_ : public fem<base_poly> {
    dim_type nc;
    mutable base_matrix K;
    base_small_vector norient;
    mutable bgeot::pgeotrans_precomp pgp;
    mutable bgeot::pgeometric_trans pgt_stored;
    // mutable pfem_precomp pfp;

    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    P1_RT0_(dim_type nc_);
  };

  void P1_RT0_::mat_trans(base_matrix &M,
                          const base_matrix &G,
                          bgeot::pgeometric_trans pgt) const {
    dim_type N = dim_type(G.nrows());
    gmm::copy(gmm::identity_matrix(), M);
    if (pgt != pgt_stored) {
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0);
      // pfp = fem_precomp(this, node_tab(0), 0);
    }
    GMM_ASSERT1(N == nc, "Sorry, this element works only in dimension " << nc);

    gmm::mult(G, pgp->grad(0), K); gmm::lu_inverse(K);
    for (unsigned i = 0; i <= nc; ++i) {
      if (!(pgt->is_linear()))
        { gmm::mult(G, pgp->grad(i), K); gmm::lu_inverse(K); }
      bgeot::base_small_vector n(nc);
      gmm::mult(gmm::transposed(K), cvr->normals()[i], n);

      M(i,i) = gmm::vect_norm2(n);
      n /= M(i,i);
      scalar_type ps = gmm::vect_sp(n, norient);
      if (ps < 0) M(i, i) *= scalar_type(-1);
      if (gmm::abs(ps) < 1E-8)
        GMM_WARNING2("RT0 : The normal orientation may be not correct");
    }
  }

  P1_RT0_::P1_RT0_(dim_type nc_) {
    nc = nc_;
    pgt_stored = 0;
    gmm::resize(K, nc, nc);
    gmm::resize(norient, nc);
    norient[0] = M_PI;
    for (unsigned i = 1; i < nc; ++i) norient[i] = norient[i-1]*M_PI;

    cvr = bgeot::simplex_of_reference(nc);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 1;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    ntarget_dim = nc;
    vtype = VECTORIAL_PRIMAL_TYPE;
    base_.resize(nc*(nc+1));


    for (size_type j = 0; j < nc; ++j)
      for (size_type i = 0; i <= nc; ++i) {
        base_[i+j*(nc+1)] = base_poly(nc, 1, short_type(j));
        if (i-1 == j) base_[i+j*(nc+1)] -= bgeot::one_poly(nc);
        if (i == 0) base_[i+j*(nc+1)] *= sqrt(opt_long_scalar_type(nc));
      }

    base_node pt(nc);
    pt.fill(scalar_type(1)/scalar_type(nc));
    add_node(normal_component_dof(nc), pt);

    for (size_type i = 0; i < nc; ++i) {
      pt[i] = scalar_type(0);
      add_node(normal_component_dof(nc), pt);
      pt[i] = scalar_type(1)/scalar_type(nc);
    }
  }

  static pfem P1_RT0(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && double(n) == params[0].num(),
                "Bad parameter");
    pfem p = std::make_shared<P1_RT0_>(dim_type(n));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        Element RT0 on parallelepideds.                                   */
  /* ******************************************************************** */

  struct P1_RT0Q_ : public fem<base_poly> {
    dim_type nc;
    mutable base_matrix K;
    base_small_vector norient;
    mutable bgeot::pgeotrans_precomp pgp;
    mutable bgeot::pgeometric_trans pgt_stored;
    // mutable pfem_precomp pfp;

    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    P1_RT0Q_(dim_type nc_);
  };

  void P1_RT0Q_::mat_trans(base_matrix &M,
                          const base_matrix &G,
                          bgeot::pgeometric_trans pgt) const {
    dim_type N = dim_type(G.nrows());
    gmm::copy(gmm::identity_matrix(), M);
    if (pgt != pgt_stored) {
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0),0);
      // pfp = fem_precomp(this, node_tab(0), 0);
    }
    GMM_ASSERT1(N == nc, "Sorry, this element works only in dimension " << nc);

    gmm::mult(G, pgp->grad(0), K); gmm::lu_inverse(K);
    for (unsigned i = 0; i < unsigned(2*nc); ++i) {
      if (!(pgt->is_linear()))
        { gmm::mult(G, pgp->grad(i), K); gmm::lu_inverse(K); }
      bgeot::base_small_vector n(nc);
      gmm::mult(gmm::transposed(K), cvr->normals()[i], n);

      M(i,i) = gmm::vect_norm2(n);
      n /= M(i,i);
      scalar_type ps = gmm::vect_sp(n, norient);
      if (ps < 0) M(i, i) *= scalar_type(-1);
      if (gmm::abs(ps) < 1E-8)
        GMM_WARNING2("RT0Q : The normal orientation may be not correct");
    }
  }

  P1_RT0Q_::P1_RT0Q_(dim_type nc_) {
    nc = nc_;
    pgt_stored = 0;
    gmm::resize(K, nc, nc);
    gmm::resize(norient, nc);
    norient[0] = M_PI;
    for (unsigned i = 1; i < nc; ++i) norient[i] = norient[i-1]*M_PI;

    cvr = bgeot::parallelepiped_of_reference(nc);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 1;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    ntarget_dim = nc;
    vtype = VECTORIAL_PRIMAL_TYPE;
    base_.resize(nc*2*nc);

    for (size_type j = 0; j < size_type(nc*2*nc); ++j)
      base_[j] = bgeot::null_poly(nc);

    for (short_type i = 0; i < nc; ++i) {
      base_[2*i+i*2*nc] = base_poly(nc, 1, i);
      base_[2*i+1+i*2*nc] = base_poly(nc, 1, i) - bgeot::one_poly(nc);
    }

    base_node pt(nc); pt.fill(0.5);

    for (size_type i = 0; i < nc; ++i) {
      pt[i] = 1.0;
      add_node(normal_component_dof(nc), pt);
      pt[i] = 0.0;
      add_node(normal_component_dof(nc), pt);
      pt[i] = 0.5;
    }
  }

  static pfem P1_RT0Q(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && double(n) == params[0].num(),
                "Bad parameter");
    pfem p = std::make_shared<P1_RT0Q_>(dim_type(n));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        Nedelec Element.                                                  */
  /* ******************************************************************** */

  struct P1_nedelec_ : public fem<base_poly> {
    dim_type nc;
    base_small_vector norient;
    std::vector<base_small_vector> tangents;
    mutable bgeot::pgeotrans_precomp pgp;
    mutable bgeot::pgeometric_trans pgt_stored;
    mutable pfem_precomp pfp;

    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    P1_nedelec_(dim_type nc_);
  };

  void P1_nedelec_::mat_trans(base_matrix &M, const base_matrix &G,
                              bgeot::pgeometric_trans pgt) const {
    bgeot::base_small_vector t(nc), v(nc);
    GMM_ASSERT1(G.nrows() == nc,
                "Sorry, this element works only in dimension " << nc);

    if (pgt != pgt_stored) {
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0);
      pfp = fem_precomp(std::make_shared<P1_nedelec_>(nc), node_tab(0), 0);
    }
    fem_interpolation_context ctx(pgp,pfp,0,G,0);

    for (unsigned i = 0; i < nb_dof(0); ++i) {
      ctx.set_ii(i);
      gmm::mult(ctx.K(), tangents[i], t);
      t /= gmm::vect_norm2(t);
      gmm::mult(gmm::transposed(ctx.B()), t, v);
      scalar_type ps = gmm::vect_sp(t, norient);
      if (ps < 0) v *= scalar_type(-1);
      if (gmm::abs(ps) < 1E-8)
        GMM_WARNING2("Nedelec element: "
                     "The normal orientation may be incorrect");

      const bgeot::base_tensor &tt = pfp->val(i);
      for (size_type j = 0; j < nb_dof(0); ++j) {
        scalar_type a = scalar_type(0);
        for (size_type k = 0; k < nc; ++k) a += tt(j, k) * v[k];
        M(j, i) = a;
      }
    }
    // In fact matrix M is diagonal (at least for linear transformations).
    // The computation can be simplified.
    gmm::lu_inverse(M);
  }

  P1_nedelec_::P1_nedelec_(dim_type nc_) {
    nc = nc_;
    pgt_stored = 0;
    gmm::resize(norient, nc);
    norient[0] = M_PI;
    for (unsigned i = 1; i < nc; ++i) norient[i] = norient[i-1]*M_PI;

    cvr = bgeot::simplex_of_reference(nc);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 1;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    ntarget_dim = nc;
    vtype = VECTORIAL_DUAL_TYPE;
    base_.resize(nc*(nc+1)*nc/2);
    tangents.resize(nc*(nc+1)*nc/2);

    std::vector<base_poly> lambda(nc+1);
    std::vector<base_vector> grad_lambda(nc+1);
    lambda[0] = bgeot::one_poly(nc);
    gmm::resize(grad_lambda[0], nc);
    gmm::fill(grad_lambda[0], scalar_type(-1));
    for (size_type i = 1; i <= nc; ++i) {
      lambda[i] = base_poly(nc, 1, short_type(i-1));
      lambda[0] -= lambda[i];
      gmm::resize(grad_lambda[i], nc);
      grad_lambda[i][i-1] = 1;
    }

    size_type j = 0;
    for (size_type k = 0; k <= nc; ++k)
      for (size_type l = k+1; l <= nc; ++l, ++j) {
        for (size_type i = 0; i < nc; ++i) {
          base_[j+i*(nc*(nc+1)/2)] = lambda[k] * grad_lambda[l][i]
            - lambda[l] * grad_lambda[k][i];
          // cout << "base(" << j << "," << i << ") = " <<  base_[j+i*(nc*(nc+1)/2)] << endl;
        }

        base_node pt = (cvr->points()[k] + cvr->points()[l]) / scalar_type(2);
        add_node(edge_component_dof(nc), pt);
        tangents[j] = cvr->points()[l] - cvr->points()[k];
        tangents[j] /= gmm::vect_norm2(tangents[j]);
        // cout << "tangent(" << j << ") = " << tangents[j] << endl;
      }
  }

  static pfem P1_nedelec(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    GMM_ASSERT1(n > 1 && n < 100 && double(n) == params[0].num(),
                "Bad parameter");
    pfem p = std::make_shared<P1_nedelec_>(dim_type(n));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        P1 element with a bubble base fonction on a face : type lagrange  */
  /* ******************************************************************** */

  struct P1_wabbfoafla_ : public PK_fem_
  { // idem elt prec mais avec raccord lagrange. A faire en dim. quelconque ..
    P1_wabbfoafla_();
  };

  P1_wabbfoafla_::P1_wabbfoafla_() : PK_fem_(2, 1) {
    unfreeze_cvs_node();
    es_degree = 2;
    base_node pt(2); pt.fill(0.5);
    add_node(lagrange_dof(2), pt);
    base_.resize(nb_dof(0));

    read_poly(base_[0], 2, "1 - y - x");
    read_poly(base_[1], 2, "x*(1 - 2*y)");
    read_poly(base_[2], 2, "y*(1 - 2*x)");
    read_poly(base_[3], 2, "4*x*y");
  }

  static pfem P1_with_bubble_on_a_face_lagrange(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = std::make_shared<P1_wabbfoafla_>();
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        PK Gauss-Lobatto element on the segment                           */
  /* ******************************************************************** */

  static const double fem_coef_gausslob_1[4] =
    { 1.000000000000000e+00, -1.000000000000000e+00, 0.000000000000000e-01,
      1.000000000000000e+00 };

  static const double fem_coef_gausslob_2[9] =
    { 1.000000000000000e+00, -3.000000000000000e+00, 2.000000000000000e+00,
      0.000000000000000e-01, 4.000000000000000e+00, -4.000000000000000e+00,
      0.000000000000000e-01, -1.000000000000000e+00, 2.000000000000000e+00 };

  static const double fem_coef_gausslob_3[16] =
    { 1.000000000000000e+00, -6.000000000000000e+00, 1.000000000000000e+01,
      -5.000000000000000e+00, 0.000000000000000e-01, 8.090169943749474e+00,
      -1.927050983124842e+01, 1.118033988749895e+01, 0.000000000000000e-01,
      -3.090169943749474e+00, 1.427050983124842e+01, -1.118033988749895e+01,
      0.000000000000000e-01, 1.000000000000000e+00, -5.000000000000000e+00,
      5.000000000000000e+00 };

  static const double fem_coef_gausslob_4[25] =
    { 1.000000000000000e+00, -1.000000000000000e+01, 3.000000000000000e+01,
      -3.500000000000000e+01, 1.400000000000000e+01, 0.000000000000000e-01,
      1.351300497744848e+01, -5.687234826567877e+01, 7.602600995489696e+01,
      -3.266666666666667e+01, 0.000000000000000e-01, -5.333333333333333e+0,
      4.266666666666667e+01, -7.466666666666667e+01, 3.733333333333333e+01,
      0.000000000000000e-01, 2.820328355884853e+00, -2.479431840098789e+01,
      5.464065671176971e+01, -3.266666666666667e+01, 0.000000000000000e-01,
      -1.000000000000000e+00, 9.000000000000000e+00, -2.100000000000000e+01,
      1.400000000000000e+01 };

  static const double fem_coef_gausslob_5[36] =
    { 1.000000000000000e+00, -1.500000000000000e+01, 7.000000000000000e+01,
      -1.400000000000000e+02, 1.260000000000000e+02, -4.200000000000000e+01,
      0.000000000000000e-01, 2.028283187263934e+01, -1.315819844629968e+02,
      2.996878573260126e+02, -2.884609153819731e+02, 1.000722106463179e+02,
      0.000000000000000e-01, -8.072374540610696e+00, 9.849823568607377e+01,
      -2.894574355386068e+02, 3.201990314777874e+02, -1.211674570846437e+02,
      0.000000000000000e-01, 4.489369296352334e+00, -6.035445290945900e+01,
      2.203358804738940e+02, -2.856382539454310e+02, 1.211674570846437e+02,
      0.000000000000000e-01, -2.699826628380976e+00, 3.743820168638206e+01,
      -1.465663022612998e+02, 2.119001378496167e+02, -1.000722106463179e+02,
      0.000000000000000e-01, 1.000000000000000e+00, -1.400000000000000e+01,
      5.600000000000000e+01, -8.400000000000000e+01, 4.200000000000000e+01 };

  static const double fem_coef_gausslob_6[49] =
    { 1.000000000000000e+00, -2.100000000000000e+01, 1.400000000000000e+02,
      -4.200000000000000e+02, 6.300000000000000e+02, -4.620000000000000e+02,
      1.320000000000000e+02, 0.000000000000000e-01, 2.840315320583963e+01,
      -2.618707992013254e+02, 8.915455952644434e+02, -1.426720320066430e+03,
      1.086906031987037e+03, -3.182636611895653e+02, 0.000000000000000e-01,
      -1.133797045109102e+01, 1.954053162272040e+02, -8.515354909154440e+02,
      1.555570646517052e+03, -1.285566162567286e+03, 3.974636611895653e+02,
      0.000000000000000e-01, 6.400000000000000e+00, -1.216000000000000e+02,
      6.528000000000000e+02, -1.382400000000000e+03, 1.267200000000000e+03,
      -4.224000000000000e+02, 0.000000000000000e-01, -4.099929626153486e+00,
      8.051601475380118e+01, -4.643586932712080e+02, 1.089694751524101e+03,
      -1.099215804570106e+03, 3.974636611895653e+02, 0.000000000000000e-01,
      2.634746871404869e+00, -5.245053177967978e+01, 3.115485889222086e+02,
      -7.661450779747230e+02, 8.226759351503546e+02, -3.182636611895653e+02,
      0.000000000000000e-01, -1.000000000000000e+00, 2.000000000000000e+01,
      -1.200000000000000e+02, 3.000000000000000e+02, -3.300000000000000e+02,
      1.320000000000000e+02 };

  static const double fem_coef_gausslob_7[64] =
    { 1.000000000000000e+00, -2.800000000000000e+01, 2.520000000000000e+02,
      -1.050000000000000e+03, 2.310000000000000e+03, -2.772000000000000e+03,
      1.716000000000000e+03, -4.290000000000000e+02, 0.000000000000000e-01,
      3.787519721423474e+01, -4.699045385400572e+02, 2.217166528901653e+03,
      -5.195916436930171e+03, 6.469992588404291e+03, -4.101225846986446e+03,
      1.042012507936495e+03, 0.000000000000000e-01, -1.513857963869697e+01,
      3.497259983590744e+02, -2.101837798737552e+03, 5.599947843186200e+03,
      -7.539551580657127e+03, 5.032695631593761e+03, -1.325841514105659e+03,
      0.000000000000000e-01, 8.595816328530350e+00, -2.189405837038675e+02,
      1.612357003153179e+03, -4.947308401212060e+03, 7.342604827965170e+03,
      -5.255204822337236e+03, 1.457896159806284e+03, 0.000000000000000e-01,
      -5.620377978515898e+00, 1.480753190084385e+02, -1.171440824431867e+03,
      3.964008996775197e+03, -6.427195249873722e+03, 4.950068296306753e+03,
      -1.457896159806284e+03, 0.000000000000000e-01, 3.883318851088245e+00,
      -1.038534676200790e+02, 8.481025943868683e+02, -3.011828579891082e+03,
      5.186049587313396e+03, -4.248194967145850e+03, 1.325841514105659e+03,
      0.000000000000000e-01, -2.595374776640464e+00, 6.989727249649080e+01,
      -5.793475032722815e+02, 2.106096578071916e+03, -3.744900173152008e+03,
      3.192861708569018e+03, -1.042012507936495e+03, 0.000000000000000e-01,
      1.000000000000000e+00, -2.700000000000000e+01, 2.250000000000000e+02,
      -8.250000000000000e+02, 1.485000000000000e+03, -1.287000000000000e+03,
      4.290000000000000e+02 };

  static const double fem_coef_gausslob_8[81] =
    { 1.000000000000000e+00, -3.600000000000000e+01, 4.200000000000000e+02,
      -2.310000000000000e+03, 6.930000000000000e+03, -1.201200000000000e+04,
      1.201200000000000e+04, -6.435000000000000e+03, 1.430000000000000e+03,
      0.000000000000000e-01, 4.869949034318613e+01, -7.815432549965997e+02,
      4.860656931912704e+03, -1.551737634456316e+04, 2.788918324236022e+04,
      -2.854121637661796e+04, 1.553203650368708e+04, -3.490440192125469e+03,
      0.000000000000000e-01, -1.947740331442309e+01, 5.805138088476765e+02,
      -4.583922431826645e+03, 1.659300238583299e+04, -3.217606831061953e+04,
      3.461498040096808e+04, -1.950464316590829e+04, 4.495614716020147e+03,
      0.000000000000000e-01, 1.108992781389876e+01, -3.644117397881921e+02,
      3.513408770397022e+03, -1.458458798126453e+04, 3.105326914181443e+04,
      -3.569574046476449e+04, 2.111700401254369e+04, -5.050031666751820e+03,
      0.000000000000000e-01, -7.314285714285714e+00, 2.486857142857143e+02,
      -2.574628571428571e+03, 1.174674285714286e+04, -2.719451428571429e+04,
      3.347017142857143e+04,  -2.091885714285714e+04, 5.229714285714286e+03,
      0.000000000000000e-01, 5.181491353118710e+00, -1.789312751409961e+02,
      1.913693930879634e+03, -9.161425477258226e+03, 2.246586272145708e+04,
      -2.927759904600966e+04, 1.928324932147088e+04, -5.050031666751820e+03,
      0.000000000000000e-01, -3.748881746893966e+00, 1.304893011815078e+02,
      -1.418925315009559e+03, 6.967886161876575e+03, -1.767073170824305e+04,
      2.395969028817416e+04, -1.646027456225289e+04, 4.495614716020147e+03,
      0.000000000000000e-01, 2.569661265399177e+00, -8.980255438911062e+01,
      9.847166850754155e+02, -4.899241601766511e+03, 1.264999919894514e+04,
      -1.754928623032154e+04, 1.239148503331668e+04, -3.490440192125469e+03,
      0.000000000000000e-01, -1.000000000000000e+00, 3.500000000000000e+01,
      -3.850000000000000e+02, 1.925000000000000e+03, -5.005000000000000e+03,
      7.007000000000000e+03, -5.005000000000000e+03, 1.430000000000000e+03 };

  static const double fem_coef_gausslob_9[100] =
    { 1.000000000000000e+00, -4.500000000000000e+01, 6.600000000000000e+02,
      -4.620000000000000e+03, 1.801800000000000e+04, -4.204200000000000e+04,
      6.006000000000000e+04, -5.148000000000000e+04, 2.431000000000000e+04,
      -4.862000000000000e+03, 0.000000000000000e-01, 6.087629005856384e+01,
      -1.226341297551814e+03, 9.697405499616728e+03, -4.021760400071670e+04,
      9.725280603247945e+04, -1.421240159771644e+05, 1.237105621496494e+05,
      -5.906188663911807e+04, 1.190819794274692e+04, 0.000000000000000e-01,
      -2.435589341485964e+01, 9.095415690555422e+02, -9.111255870205338e+03,
      4.276661412893713e+04, -1.114146601604227e+05, 1.709573510663859e+05,
      -1.539309822784481e+05, 7.531473186776967e+04, -1.546698442965720e+04,
      0.000000000000000e-01, 1.388757697026791e+01, -5.717395054991898e+02,
      6.975542885191629e+03, -3.743822964286359e+04, 1.068054892026842e+05,
      -1.747039036097183e+05, 1.648204809285228e+05, -8.352714678107793e+04,
      1.762561894579012e+04, 0.000000000000000e-01, -9.198709522206266e+00,
      3.919017281073292e+02, -5.132147808492991e+03, 3.020136030457121e+04,
      -9.337958558844686e+04, 1.629937209113228e+05, -1.619399017586618e+05,
      8.553993533073839e+04, -1.866608440961585e+04, 0.000000000000000e-01,
      6.589286067498368e+00, -2.852085019651226e+02, 3.859417687766574e+03,
      -2.381847798089535e+04, 7.784545414265617e+04, -1.434184925463668e+05,
      1.495994578589254e+05, -8.245482435580428e+04, 1.866608440961585e+04,
      0.000000000000000e-01, -4.905768350885374e+00, 2.141208512030290e+02,
      -2.948048350518040e+03, 1.867520721718195e+04, -6.311993447254502e+04,
      1.208313444661297e+05, -1.311255887283438e+05, 7.510342373103317e+04,
      -1.762561894579012e+04, 0.000000000000000e-01, 3.659127863806492e+00,
      -1.604518938956052e+02, 2.230466872754075e+03, -1.433960781600324e+04,
      4.943623515122416e+04, -9.697372467640532e+04, 1.082245668039501e+05,
      -6.388812799914516e+04, 1.546698442965720e+04, 0.000000000000000e-01,
      -2.551909672185327e+00, 1.121770505458310e+02, -1.567380916112636e+03,
      1.015673778978859e+04, -3.539780430762936e+04, 7.040572036581639e+04,
      -7.991059497559391e+04, 4.811189484560421e+04, -1.190819794274692e+04,
      0.000000000000000e-01, 1.000000000000000e+00, -4.400000000000000e+01,
      6.160000000000000e+02, -4.004000000000000e+03, 1.401400000000000e+04,
      -2.802800000000000e+04, 3.203200000000000e+04, -1.944800000000000e+04,
      4.862000000000000e+03 };

  static const double fem_coef_gausslob_10[121] =
    { 1.000000000000000e+00, -5.500000000000000e+01, 9.900000000000000e+02,
      -8.580000000000000e+03, 4.204200000000000e+04, -1.261260000000000e+05,
      2.402400000000000e+05, -2.917200000000000e+05, 2.187900000000000e+05,
      -9.237800000000000e+04, 1.679600000000000e+04, 0.000000000000000e-01,
      7.440573476392379e+01, -1.837547309495916e+03, 1.797721877249297e+04,
      -9.362519219895049e+04, 2.909773746534557e+05, -5.668103968059107e+05,
      6.987888266998443e+05, -5.297630128145374e+05, 2.254581472606030e+05,
      -4.123982399226536e+04, 0.000000000000000e-01, -2.977479259019735e+01,
      1.361302600480045e+03, -1.684411461834327e+04, 9.915381814787320e+04,
      -3.316413447923031e+05, 6.777335995676564e+05, -8.637075009663581e+05,
      6.706702925312933e+05, -2.905859066780586e+05, 5.388962900035043e+04,
      0.000000000000000e-01, 1.699123898926703e+01, -8.563552206749944e+02,
      1.288193007592445e+04, -8.652550760766087e+04, 3.163119148667456e+05,
      -6.879421746683590e+05, 9.173107022760963e+05, -7.368809317678005e+05,
      3.277210563158369e+05, -6.203762550909711e+04, 0.000000000000000e-01,
      -1.128077599537177e+01, 5.884060270969327e+02, -9.496934299975062e+03,
      6.981839702203256e+04, -2.759867860090975e+05, 6.390150586777585e+05,
      -8.953334100127128e+05, 7.481407085083284e+05, -3.434511859876541e+05,
      6.671702685021839e+04, 0.000000000000000e-01, 8.126984126984127e+00,
      -4.307301587301587e+02, 7.184253968253968e+03, -5.536101587301587e+04,
      2.309526349206349e+05, -5.631187301587302e+05, 8.261892063492063e+05,
      -7.184253968253968e+05, 3.412520634920635e+05, -6.825041269841270e+04,
      0.000000000000000e-01, -6.131078401763724e+00, 3.277460052754944e+02,
      -5.563892337052540e+03, 4.401679638240306e+04, -1.898229640674875e+05,
      4.802970424048780e+05, -7.315927845245721e+05, 6.593462428792687e+05,
      -3.237190825145298e+05, 6.671702685021839e+04, 0.000000000000000e-01,
      4.719539826177156e+00, -2.535442364309725e+02, 4.348374949258725e+03,
      -3.493745849693315e+04, 1.537770968392435e+05, -3.987659746141970e+05,
      6.242937855878344e+05, -5.790845728346388e+05, 2.926551987751342e+05,
      -6.203762550909711e+04, 0.000000000000000e-01, -3.595976071589556e+00,
      1.937484128537626e+02, -3.342868418261306e+03, 2.710687970739982e+04,
      -1.208013807254569e+05, 3.181552127960248e+05, -5.073176789159280e+05,
      4.804304374445346e+05, -2.483103833254456e+05, 5.388962900035043e+04,
      0.000000000000000e-01, 2.539125352570295e+00, -1.370261203741934e+02,
      2.372031907702074e+03, -1.933271708314826e+04, 8.675745431426523e+04,
      -2.305316371991209e+05, 3.716008535065897e+05, -3.564317671210514e+05,
      1.869400926620506e+05, -4.123982399226536e+04, 0.000000000000000e-01,
      -1.000000000000000e+00, 5.400000000000000e+01, -9.360000000000000e+02,
      7.644000000000000e+03, -3.439800000000000e+04, 9.172800000000000e+04,
      -1.485120000000000e+05, 1.432080000000000e+05, -7.558200000000000e+04,
      1.679600000000000e+04 };

  static const double fem_coef_gausslob_11[144] =
    { 1.000000000000000e+00, -6.600000000000000e+01, 1.430000000000000e+03,
      -1.501500000000000e+04, 9.009000000000000e+04, -3.363360000000000e+05,
      8.168160000000000e+05, -1.312740000000000e+06, 1.385670000000000e+06,
      -9.237800000000000e+05, 3.527160000000000e+05, -5.878600000000000e+04,
      0.000000000000000e-01, 8.928790437897459e+01, -2.652104227906408e+03,
      3.141784850363868e+04, -2.002791716233576e+05, 7.743819221375186e+05,
      -1.922871126014229e+06, 3.137025618338122e+06, -3.346678943928588e+06,
      2.248625248986712e+06, -8.636670995581690e+05, 1.446085194818801e+05,
      0.000000000000000e-01, -3.573451504477809e+01, 1.963011183403142e+03,
      -2.937610003978132e+04, 2.114542549822465e+05, -8.792000471014727e+05,
      2.288870840565664e+06, -3.858042797257275e+06, 4.213930780912515e+06,
      -2.881507044264936e+06, 1.121761824282757e+06, -1.898189887480762e+05,
      0.000000000000000e-01, 2.040222049137883e+01,  -1.235400296692680e+03,
      2.244501976036562e+04, -1.840644193090062e+05, 8.352985709834925e+05,
      -2.311501023251837e+06, 4.072373742578464e+06, -4.597525083798586e+06,
      3.224564284996090e+06, -1.280533828091591e+06, 2.201577342088092e+05,
      0.000000000000000e-01, -1.356395972416294e+01, 8.500434612026409e+02,
      -1.656519758544980e+04, 1.484886632288921e+05, -7.274015626850215e+05,
      2.139270125221522e+06, -3.953929242730888e+06, 4.636483776329532e+06,
      -3.352298848558998e+06, 1.364514095203156e+06, -2.393982879242230e+05,
      0.000000000000000e-01, 9.803369220897211e+00, -6.243148521745071e+02,
      1.257271925920887e+04, -1.180754347844014e+05, 6.096877374094766e+05,
      -1.885008008173690e+06, 3.641310114569300e+06, -4.434918589444567e+06,
      3.311645240511804e+06, -1.385401912847929e+06, 2.488026449837513e+05,
      0.000000000000000e-01, -7.447686911212630e+00, 4.784415903433910e+02,
      -9.808275331323619e+03, 9.456732952354300e+04, -5.045513347291155e+05,
      1.617062776783016e+06, -3.237833360324115e+06, 4.079238919323811e+06,
      -3.141771586138831e+06, 1.351427181973335e+06, -2.488026449837513e+05,
      0.000000000000000e-01, 5.819619912607334e+00, -3.757783851118444e+02,
      7.785050290867036e+03, -7.625636515182549e+04, 4.153153824802320e+05,
      -1.363841143951918e+06, 2.804561170833446e+06, -3.631789084056234e+06,
      2.874063732359706e+06, -1.268867071963298e+06, 2.393982879242230e+05,
      0.000000000000000e-01, -4.587084978600364e+00, 2.971291972095372e+02,
      -6.195597982053585e+03, 6.128651035627590e+04, -3.381847677955126e+05,
      1.128582073344179e+06, -2.364480249965060e+06, 3.125557361498120e+06,
      -2.527901385564680e+06, 1.141201248205309e+06, -2.201577342088092e+05,
      0.000000000000000e-01, 3.549738472143437e+00, -2.303803824096343e+02,
      4.822860472410032e+03, -4.799737703690675e+04, 2.670306747478879e+05,
      -9.003482951717774e+05, 1.909697516429202e+06, -2.560483668180433e+06,
      2.103933182581560e+06, -9.662470519460815e+05, 1.898189887480762e+05,
      0.000000000000000e-01, -2.529605817247381e+00, 1.643527121363626e+02,
      -3.448327347881922e+03, 3.443600981453996e+04, -1.924805754474854e+05,
      6.528637806490704e+05, -1.394862512471197e+06, 1.886334531344429e+06,
      -1.565422824908427e+06, 7.270266147425120e+05, -1.446085194818801e+05,
      0.000000000000000e-01, 1.000000000000000e+00, -6.500000000000000e+01,
      1.365000000000000e+03, -1.365000000000000e+04, 7.644000000000000e+04,
      -2.598960000000000e+05, 5.569200000000000e+05, -7.558200000000000e+05,
      6.298500000000000e+05, -2.939300000000000e+05, 5.878600000000000e+04 };

  static const double fem_coef_gausslob_12[169] =
    { 1.000000000000000e+00, -7.800000000000000e+01, 2.002000000000000e+03,
      -2.502500000000000e+04, 1.801800000000000e+05, -8.168160000000000e+05,
      2.450448000000000e+06, -4.988412000000000e+06, 6.928350000000000e+06,
      -6.466460000000000e+06, 3.879876000000000e+06, -1.352078000000000e+06,
      2.080120000000000e+05, 0.000000000000000e-01, 1.055228477237708e+02,
      -3.710649283521791e+03, 5.230891096204477e+04, -4.000264992878070e+05,
      1.877737871587605e+06, -5.758751500257082e+06, 1.189876582421287e+07,
      -1.670085336595664e+07, 1.570840983751716e+07, -9.480360124877962e+06,
      3.318799039873028e+06, -5.124248673374161e+05, 0.000000000000000e-01,
      -4.223530748650643e+01, 2.744602756239140e+03, -4.883026612748412e+04,
      4.213447894212839e+05, -2.125569662252139e+06, 6.831231541213443e+06,
      -1.457745185889376e+07, 2.094131826427298e+07, -2.004063010887189e+07,
      1.225627209853367e+07, -4.335340118801754e+06, 6.749529540569050e+05,
      0.000000000000000e-01, 2.412126940135046e+01, -1.727728082317750e+03,
      3.727953470399318e+04, -3.660428937429080e+05, 2.013286677665669e+06,
      -6.871455230919223e+06, 1.531439984708661e+07, -2.272429867457359e+07,
      2.229291249432404e+07, -1.390087225188044e+07, 4.993770899110666e+06,
      -7.872767949619026e+05, 0.000000000000000e-01, -1.605014152757175e+01,
      1.189832345018574e+03, -2.753035300171167e+04, 2.951729666043859e+05,
      -1.750245295939125e+06, 6.340418364700963e+06, -1.480658414760223e+07,
      2.279585234765233e+07, -2.303126179585568e+07, 1.470734589654657e+07,
      -5.387526099148482e+06, 8.631843338394749e+05, 0.000000000000000e-01,
      1.162284069412542e+01, -8.756167723813171e+02, 2.093616690467957e+04,
      -2.350848402511507e+05, 1.467905987404880e+06, -5.583024412516426e+06,
      1.360724316266146e+07, -2.172800271110029e+07, 2.264080373496021e+07,
      -1.484050586374976e+07, 5.558088637639386e+06, -9.074958680213037e+05,
      0.000000000000000e-01, -8.865800865800866e+00, 6.738008658008658e+02,
      -1.640173160173160e+04, 1.890632034632035e+05, -1.219313593073593e+06,
      4.803100813852814e+06, -1.211898237229437e+07, 1.998830268398268e+07,
      -2.144876606060606e+07, 1.443281454545455e+07, -5.532578909090909e+06,
      9.220964848484848e+05, 0.000000000000000e-01, 6.984318980773296e+00,
      -5.335955916987455e+02, 1.312836634638491e+04, -1.537652068533838e+05,
      1.012269836848560e+06, -4.084347297774684e+06, 1.057602476941974e+07,
      -1.790936242524428e+07, 1.971847079705798e+07, -1.359625813912256e+07,
      5.331861778616258e+06, -9.074958680213037e+05, 0.000000000000000e-01,
      -5.596679201904262e+00, 4.289927383042951e+02, -1.062596939367885e+04,
      1.257256555151274e+05, -8.388435073376655e+05, 3.440109149730765e+06,
      -9.074697250266149e+06, 1.567950042058767e+07, -1.762881516112804e+07,
      1.241472483931862e+07, -4.970685906925217e+06, 8.631843338394749e+05,
      0.000000000000000e-01, 4.489137849846207e+00, -3.448281543178960e+02,
      8.578250876817155e+03, -1.021659510074640e+05, 6.876731048919487e+05,
      -2.851145716716003e+06, 7.618634882796075e+06, -1.335715271315875e+07,
      1.525930546501226e+07, -1.092966082914868e+07, 4.453550640432165e+06,
      -7.872767949619026e+05, 0.000000000000000e-01, -3.514808358523940e+00,
      2.703477424140389e+02, -6.743800341396192e+03, 8.065306199056715e+04,
      -5.459331868312813e+05, 2.279586137602263e+06, -6.143562568432354e+06,
      1.087848437431968e+07, -1.256803423488744e+07, 9.114425759470104e+06,
      -3.764095329881106e+06, 6.749529540569050e+05, 0.000000000000000e-01,
      2.522322790441051e+00, -1.941585635394145e+02, 4.850890672082830e+03,
      -5.815428585185421e+04, 3.949277670351431e+05, -1.655905848916830e+06,
      4.485333711312109e+06, -7.989838200781784e+06, 9.294715032477446e+06,
      -6.793611930544113e+06, 2.830299368175965e+06, -5.124248673374161e+05,
      0.000000000000000e-01, -1.000000000000000e+00, 7.700000000000000e+01,
      -1.925000000000000e+03, 2.310000000000000e+04, -1.570800000000000e+05,
      6.597360000000000e+05, -1.790712000000000e+06, 3.197700000000000e+06,
      -3.730650000000000e+06, 2.735810000000000e+06, -1.144066000000000e+06,
      2.080120000000000e+05 };

  static const double fem_coef_gausslob_13[196] =
    { 1.000000000000000e+00, -9.100000000000000e+01, 2.730000000000000e+03,
      -4.004000000000000e+04, 3.403400000000000e+05, -1.837836000000000e+06,
      6.651216000000000e+06, -1.662804000000000e+07, 2.909907000000000e+07,
      -3.556553000000000e+07, 2.974571600000000e+07, -1.622493600000000e+07,
      5.200300000000000e+06, -7.429000000000000e+05, 0.000000000000000e-01,
      1.231105960095249e+02, -5.057514000718615e+03, 8.362619816879477e+04,
      -7.548172444491492e+05, 4.219784854542762e+06, -1.560990588170226e+07,
      3.960523865471669e+07, -7.003645336672149e+07, 8.625846242015506e+07,
      -7.256273938600024e+07, 3.975792117062384e+07, -1.278833059440045e+07,
      1.832147578471145e+06, 0.000000000000000e-01, -4.927732466948325e+01,
      3.738734011094227e+03, -7.796486012083982e+04, 7.935560370804071e+05,
      -4.765562636392240e+06, 1.846680880578437e+07, -4.837506802694851e+07,
      8.753277424234758e+07, -1.096660444159086e+08, 9.346798694406962e+07,
      -5.173889595224656e+07, 1.677849814552107e+07, -2.419777739872722e+06,
      0.000000000000000e-01, 2.814884111282567e+01, -2.353904695653627e+03,
      5.948276501192433e+04, -6.883051529064705e+05, 4.502895601024915e+06,
      -1.851735708867110e+07, 5.063079101609929e+07, -9.458217148027056e+07,
      1.214199817163488e+08, -1.054743067017811e+08, 5.927651305189011e+07,
      -1.946011726944643e+07, 2.834919298555227e+06, 0.000000000000000e-01,
      -1.874042032746388e+01, 1.621968976936385e+03, -4.394233902947216e+04,
      5.547892506224127e+05, -3.908876121817967e+06, 1.704431633766850e+07,
      -4.878627691196220e+07, 9.448003328782282e+07, -1.248200449226441e+08,
      1.109678546438889e+08, -6.355498649510845e+07, 2.119358718443647e+07,
      -3.128057142433586e+06, 0.000000000000000e-01, 1.358783086373605e+01,
      -1.195146716951513e+03, 3.345811195278515e+04, -4.422483366082136e+05,
      3.278781781106079e+06, -1.499532485039955e+07, 4.474689669452104e+07,
      -8.978037194074775e+07, 1.222039761714010e+08, -1.114085938050346e+08,
      6.517880226738924e+07, -2.213159774622623e+07, 3.317403211532315e+06,
      0.000000000000000e-01, -1.039065134180050e+01, 9.220321824274881e+02,
      -2.627964906979335e+04, 3.565631308351814e+05, -2.729347397416285e+06,
      1.291899991743804e+07, -3.987098284679090e+07, 8.253644504125924e+07,
      -1.155541226713629e+08, 1.080161713852185e+08, -6.460510650895925e+07,
      2.236736005147009e+07, -3.410612094153076e+06, 0.000000000000000e-01,
      8.225051804237637e+00, -7.337438600306013e+02, 2.113982918743374e+04,
      -2.914573383452468e+05, 2.277144431399071e+06, -1.103660550091489e+07,
      3.493361321847434e+07, -7.418006034176242e+07, 1.064417028079657e+08,
      -1.018292957440870e+08, 6.222452923525811e+07, -2.197059717251990e+07,
      3.410612094153076e+06, 0.000000000000000e-01,  -6.651370533890156e+00,
      5.953674398464600e+02, -1.727143617692029e+04, 2.405949101477657e+05,
      -1.905359074321061e+06, 9.386078020968711e+06, -3.025905095154839e+07,
      6.552811535463406e+07, -9.594395490329804e+07, 9.365009838355809e+07,
      -5.835707981219508e+07, 2.099464400369387e+07, -3.317403211532315e+06,
      0.000000000000000e-01, 5.430796129527214e+00, -4.871978584294892e+02,
      1.419769025501944e+04, -1.991370307462581e+05, 1.591472100521582e+06,
      -7.928247009292726e+06, 2.589562028603176e+07, -5.690356974983853e+07,
      8.463743197871011e+07, -8.398458536550263e+07, 5.322039739169053e+07,
      -1.947115566720015e+07, 3.128057142433586e+06, 0.000000000000000e-01,
      -4.414467779036191e+00, 3.966097971621681e+02, -1.159268854505275e+04,
      1.633445673328560e+05, -1.313458735263122e+06, 6.593624701202045e+06,
      -2.173390279168494e+07, 4.826160481317836e+07, -7.262663174126566e+07,
      7.298651647234048e+07, -4.687881110584063e+07, 1.739383361177152e+07,
      -2.834919298555227e+06, 0.000000000000000e-01, 3.487743182287134e+00,
      -3.136500314357888e+02, 9.185689380767778e+03, -1.298134042763866e+05,
      1.048017196494400e+06, -5.287706376855868e+06, 1.753577438278919e+07,
      -3.921741432164137e+07, 5.949694434313328e+07, -6.033542452985016e+07,
      3.913958191606598e+07, -1.467861247282431e+07, 2.419777739872722e+06,
      0.000000000000000e-01, -2.516624450464659e+00, 2.264447557529062e+02,
      -6.639311014646825e+03, 9.399061131310191e+04, -7.605959998781342e+05,
      3.848998924774717e+06, -1.281093272369737e+07, 2.877371846174006e+07,
      -4.386952078323465e+07, 4.473878170318014e+07, -2.920546515856783e+07,
      1.102958792572444e+07, -1.832147578471145e+06, 0.000000000000000e-01,
      1.000000000000000e+00, -9.000000000000000e+01, 2.640000000000000e+03,
      -3.740000000000000e+04, 3.029400000000000e+05, -1.534896000000000e+06,
      5.116320000000000e+06, -1.151172000000000e+07, 1.758735000000000e+07,
      -1.797818000000000e+07, 1.176753600000000e+07, -4.457400000000000e+06,
      7.429000000000000e+05 };

  static const double fem_coef_gausslob_14[225] =
    { 1.000000000000000e+00, -1.050000000000000e+02, 3.640000000000000e+03,
      -6.188000000000000e+04, 6.126120000000000e+05, -3.879876000000000e+06,
      1.662804000000000e+07, -4.988412000000000e+07, 1.066965900000000e+08,
      -1.636014380000000e+08, 1.784742960000000e+08, -1.352078000000000e+08,
      6.760390000000000e+07, -2.005830000000000e+07, 2.674440000000000e+06,
      0.000000000000000e-01, 1.420511699552723e+02, -6.740724197514907e+03,
      1.291563810650317e+05, -1.357536885890637e+06, 8.899789739175488e+06,
      -3.898284725729828e+07, 1.186784002318284e+08, -2.564866709310747e+08,
      3.962828733570607e+08, -4.348015522628213e+08, 3.308658899677545e+08,
      -1.660170000478420e+08, 4.939776003229131e+07, -6.601663651220939e+06,
      0.000000000000000e-01, -5.686066782897496e+01, 4.980782943180571e+03,
      -1.202886759366057e+05, 1.425067611910860e+06, -1.003204881513823e+07,
      4.601736532886305e+07, -1.446080878593359e+08, 3.197256124417587e+08,
      -5.024241189222256e+08, 5.584380089699472e+08, -4.292685524711295e+08,
      2.171358326952990e+08, -6.503152653250142e+07, 8.737812306213077e+06,
      0.000000000000000e-01, 3.248522677540712e+01, -3.136209432322820e+03,
      9.172216152088957e+04, -1.234458149638161e+06, 9.460578149039340e+06,
      -4.602710110035357e+07, 1.508977142152744e+08, -3.443000936718750e+08,
      5.541917935770469e+08, -6.276282328622294e+08, 4.896977188070307e+08,
      -2.507043130148833e+08, 7.583045543918066e+07, -1.027267982590785e+07,
      0.000000000000000e-01, -2.163547691954172e+01, 2.161829695966707e+03,
      -6.777232432848026e+04, 9.945601317987818e+05, -8.202377657972944e+06,
      4.227975782999525e+07, -1.449995018490026e+08, 3.427553248466447e+08,
      -5.674380093336526e+08, 6.573468625138306e+08, -5.224446315267705e+08,
      2.715766154508912e+08, -8.319461131269656e+07, 1.139164303704407e+07,
      0.000000000000000e-01, 1.569978564006007e+01, -1.594280678736617e+03,
      5.164364569735884e+04, -7.932250762767995e+05, 6.879605840139093e+06,
      -3.716431671531196e+07, 1.327627119052969e+08, -3.248633554644047e+08,
      5.536614212284358e+08, -6.572276042331931e+08, 5.332102141003966e+08,
      -2.820526436178851e+08, 8.770029066945359e+07, -1.216316370145453e+07,
      0.000000000000000e-01, -1.202518395631915e+01, 1.231993083463710e+03,
      -4.063141778456666e+04, 6.405521495953326e+05, -5.734055805513453e+06,
      3.204057307537486e+07, -1.182863821502541e+08, 2.983631937198568e+08,
      -5.225422090403253e+08, 6.354190974001179e+08, -5.265537919663995e+08,
      2.837551732890861e+08, -8.968009595208465e+07, 1.261735673043107e+07,
      0.000000000000000e-01, 9.547785547785548e+00, -9.834219114219114e+02,
      3.278709557109557e+04, -5.252427785547786e+05, 4.798602442890443e+06,
      -2.744701911421911e+07, 1.038669217715618e+08, -2.685490364568765e+08,
      4.816180870862471e+08, -5.987952711608392e+08, 5.064437616783217e+08,
      -2.780475554312354e+08, 8.937242853146853e+07, -1.276748979020979e+07,
      0.000000000000000e-01, -7.763592643695499e+00, 8.024013730981778e+02,
      -2.693903659000494e+04, 4.360799342555947e+05, -4.038452021764897e+06,
      2.347605516322762e+07, -9.046087127754393e+07, 2.384165701554799e+08,
      -4.360078989902932e+08, 5.526354677146948e+08, -4.761786531169400e+08,
      2.660933883812130e+08, -8.696289827395033e+07, 1.261735673043107e+07,
      0.000000000000000e-01, 6.402657115106499e+00, -6.632652203626004e+02,
      2.237191510124190e+04, -3.647008347515229e+05, 3.408912135873189e+06,
      -2.004238739169268e+07, 7.824760241311718e+07, -2.092325230710293e+08,
      3.885803431690498e+08, -5.004334616015021e+08, 4.381904244262927e+08,
      -2.487967617473505e+08, 8.258400115090981e+07, -1.216316370145453e+07,
      0.000000000000000e-01, -5.303584550798654e+00, 5.502727059685037e+02,
      -1.861988467344103e+04, 3.050015660700474e+05, -2.869271809771707e+06,
      1.700462338220501e+07, -6.701518639186630e+07, 1.811217810430339e+08,
      -3.403535526125257e+08, 4.438883801280693e+08, -3.938531369776322e+08,
      2.266861847568459e+08, -7.628839120592035e+07, 1.139164303704407e+07,
      0.000000000000000e-01, 4.356127432886088e+00, -4.524531153029460e+02,
      1.534317874813675e+04, -2.521565333504139e+05, 2.382646312619400e+06,
      -1.419908547341188e+07, 5.633074020272954e+07, -1.534171364617007e+08,
      2.907942363862238e+08, -3.828802350952767e+08, 3.432339697459343e+08,
      -1.997222564631490e+08, 6.798706212352923e+07, -1.027267982590785e+07,
      0.000000000000000e-01, -3.466327490120249e+00, 3.602867454806401e+02,
      -1.223518159744950e+04, 2.015152850494290e+05, -1.909713800970267e+06,
      1.142278748458884e+07, -4.551909122318173e+07, 1.246206804545239e+08,
      -2.376275441309645e+08, 3.149824199011395e+08, -2.844660497989077e+08,
      1.668669076381706e+08, -5.729784575448167e+07, 8.737812306213077e+06,
      0.000000000000000e-01, 2.512080922932578e+00, -2.612119914965073e+02,
      8.878143206793750e+03, -1.464124202177336e+05, 1.389929291394544e+06,
      -8.332053211967151e+06, 3.329158201137647e+07, -9.143262460433713e+07,
      1.749809182259230e+08, -2.329047114119376e+08, 2.113183971320489e+08,
      -1.245975118891604e+08, 4.302553108480184e+07, -6.601663651220939e+06,
      0.000000000000000e-01, -1.000000000000000e+00, 1.040000000000000e+02,
      -3.536000000000000e+03, 5.834400000000000e+04, -5.542680000000000e+05,
      3.325608000000000e+06, -1.330243200000000e+07, 3.658168800000000e+07,
      -7.011490200000000e+07, 9.348653600000000e+07, -8.498776000000000e+07,
      5.022004000000000e+07, -1.738386000000000e+07, 2.674440000000000e+06 };

  static const double fem_coef_gausslob_16[289] =
    { 1.000000000000000e+00, -1.360000000000000e+02, 6.120000000000000e+03,
      -1.356600000000000e+05, 1.763580000000000e+06, -1.481407200000000e+07,
      8.535727200000000e+07, -3.505745100000000e+08, 1.051723530000000e+09,
      -2.337163400000000e+09, 3.866943080000000e+09, -4.745793780000000e+09,
      4.259045700000000e+09, -2.714556600000000e+09, 1.163381400000000e+09,
      -3.005401950000000e+08, 3.535767000000000e+07, 0.000000000000000e-01,
      1.839908474110340e+02, -1.132675577023645e+04, 2.828774748172428e+05,
      -3.903228397113182e+06, 3.393217989215092e+07, -1.997936813387011e+08,
      8.326183533203730e+08, -2.523654441487500e+09, 5.650496513849965e+09,
      -9.402288564814163e+09, 1.159004125992500e+10, -1.043753556167033e+10,
      6.671119113102151e+09, -2.865585732582308e+09, 7.416762022559170e+08,
      -8.739414676532781e+07, 0.000000000000000e-01, -7.365158560983221e+01,
      8.363752486870456e+03, -2.630512922725150e+05, 4.088268892503375e+06,
      -3.814296099037553e+07, 2.350888858038512e+08, -1.010915772862718e+09,
      3.133749897384450e+09, -7.134585101325640e+09, 1.202392366962845e+10,
      -1.496979454364896e+10, 1.358833701849443e+10, -8.740756338653663e+09,
      3.774397412355719e+09, -9.811765345365573e+08, 1.160408606498937e+08,
      0.000000000000000e-01, 4.208515410469884e+01, -5.266887544418802e+03,
      2.004067164817591e+05, -3.534528216742270e+06, 3.586506864599599e+07,
      -2.342572871648561e+08, 1.050195570375994e+09, -3.357626237614668e+09,
      7.826157982970905e+09, -1.343314504492630e+10, 1.696909064108448e+10,
      -1.558480944857237e+10, 1.012167818105129e+10, -4.405611470443590e+09,
      1.152926420144654e+09, -1.371250292488943e+08, 0.000000000000000e-01,
      -2.804154671449467e+01, 3.632134644860107e+03, -1.481030987143817e+05,
      2.845430205439061e+06, -3.103475950537538e+07, 2.145184209516461e+08,
      -1.004950951655481e+09, 3.325504487572721e+09, -7.965634686855131e+09,
      1.397533464514550e+10, -1.797134014690632e+10, 1.674913384367476e+10,
      -1.101142723314074e+10, 4.842306972881947e+09, -1.278281396603255e+09,
      1.531698732399162e+08, 0.000000000000000e-01, 2.036791285538472e+01,
      -2.681212491930484e+03, 1.129589658306899e+05, -2.270501508829428e+06,
      2.601887714173907e+07, -1.882644410377163e+08, 9.175357517098464e+08,
      -3.139134215049928e+09, 7.731773974122787e+09, -1.388518223176127e+10,
      1.820883321323428e+10, -1.725391802961138e+10, 1.150422262078233e+10,
      -5.120395806896533e+09, 1.365808881323657e+09, -1.651383905702324e+08,
      0.000000000000000e-01, -1.562975981216744e+01, 2.075857199589987e+03,
      -8.904128307330524e+04, 1.836683464233000e+06, -2.171339625537650e+07,
      1.623702309776589e+08, -8.168673425840398e+08, 2.877184244039678e+09,
      -7.272633187920932e+09, 1.336161532561319e+10, -1.787465369706492e+10,
      1.723415209556249e+10, -1.166677727073840e+10, 5.262202876034756e+09,
      -1.420107794637259e+09, 1.734782145645626e+08, 0.000000000000000e-01,
      1.245158740600359e+01, -1.662689739514946e+03, 7.210078018619273e+04,
      -1.511262926531350e+06, 1.823010400853032e+07, -1.394732137018548e+08,
      7.186625912313578e+08, -2.591802100670653e+09, 6.699969496687534e+09,
      -1.256822106464210e+10, 1.713562111741701e+10, -1.680796707246793e+10,
      1.155571599892056e+10, -5.285087289024682e+09, 1.444204616664480e+09,
      -1.784123720377501e+08, 0.000000000000000e-01, -1.018430458430458e+01,
      1.364696814296814e+03, -5.959855042735043e+04, 1.262405659052059e+06,
      -1.543602456068376e+07, 1.199989722604507e+08, -6.293065120124320e+08,
      2.311744565308469e+09, -6.087583637383061e+09, 1.162721665412277e+10,
      -1.612769282864336e+10, 1.607722369253147e+10, -1.122097126220979e+10,
      5.203928701314685e+09, -1.440373122685315e+09, 1.800466403356643e+08,
      0.000000000000000e-01, 8.484036081962663e+00, -1.139564172918365e+03,
      5.000628114583172e+04, -1.066865683702600e+06, 1.316848914076596e+07,
      -1.035421266853741e+08, 5.500823983897466e+08, -2.049399257475671e+09,
      5.477078740096760e+09, -1.061962761323504e+10, 1.495184835525608e+10,
      -1.512401891411349e+10, 1.070494963879464e+10, -5.031502683587495e+09,
      1.410393335939522e+09, -1.784123720377501e+08, 0.000000000000000e-01,
      -7.151250283691663e+00, 9.621468006776697e+02, -4.236328367402523e+04,
      9.083924059514159e+05, -1.128778312795541e+07, 8.948673396532559e+07,
      -4.799806642461592e+08, 1.807454740270899e+09, -4.886699456126079e+09,
      9.591077381959552e+09, -1.367409274689175e+10, 1.400781324267719e+10,
      -1.004054471299105e+10, 4.777971704223384e+09, -1.355543638395742e+09,
      1.734782145645626e+08, 0.000000000000000e-01, 6.060147075943005e+00,
      -8.163167553056976e+02, 3.602890134025652e+04, -7.753708269797396e+05,
      9.681484150831325e+06, -7.721340002337801e+07, 4.170906086685726e+08,
      -1.583343835764609e+09, 4.319156776154927e+09, -8.559301071696637e+09,
      1.232825943540154e+10, -1.276387222258454e+10, 9.248884856115279e+09,
      -4.449869455469566e+09, 1.276405367800062e+09, -1.651383905702324e+08,
      0.000000000000000e-01, -5.123522643555085e+00, 6.907394286218810e+02,
      -3.053901287681582e+04, 6.589382296622791e+05, -8.256408016568484e+06,
      6.613528180437879e+07, -3.591109349416936e+08, 1.371451685008549e+09,
      -3.766497172573159e+09, 7.519828793959344e+09, -1.091857986975193e+10,
      1.140164818726860e+10, -8.336452758217789e+09, 4.048470812623064e+09,
      -1.172436575235404e+09, 1.531698732399162e+08, 0.000000000000000e-01,
      4.271888353674923e+00, -5.762713061877649e+02, 2.550919052304026e+04,
      -5.514258520673562e+05, 6.926418073801134e+06, -5.565457178175802e+07,
      3.033329010654617e+08, -1.163492208450654e+09, 3.211252052317023e+09,
      -6.446888262886903e+09, 9.417864122967573e+09, -9.899668972442366e+09,
      7.289624669351123e+09, -3.566718678141100e+09, 1.041074047837655e+09,
      -1.371250292488943e+08, 0.000000000000000e-01, -3.434977405385288e+00,
      4.635617485626016e+02, -2.053688028669370e+04, 4.444943513906060e+05,
      -5.592632684586483e+06, 4.503253959356561e+07, -2.460675245081598e+08,
      9.466718497394927e+08, -2.621823641133609e+09, 5.284002694315866e+09,
      -7.752423037115038e+09, 8.187712309040204e+09, -6.060153271928369e+09,
      2.981652672294607e+09, -8.754772358617425e+08, 1.160408606498937e+08,
      0.000000000000000e-01, 2.505373764729175e+00, -3.381913429670035e+02,
      1.499009100007397e+04, -3.246847962658711e+05, 4.089321087106837e+06,
      -3.296978262323839e+07, 1.804331430493320e+08, -6.954301078105766e+08,
      1.930060872117711e+09, -3.899125665782255e+09, 5.735918309736332e+09,
      -6.075963842786729e+09, 4.511802094762441e+09, -2.227740310582889e+09,
      6.566301459893279e+08, -8.739414676532781e+07, 0.000000000000000e-01,
      -1.000000000000000e+00, 1.350000000000000e+02, -5.985000000000000e+03,
      1.296750000000000e+05, -1.633905000000000e+06, 1.318016700000000e+07,
      -7.217710500000000e+07, 2.783974050000000e+08, -7.733261250000000e+08,
      1.563837275000000e+09, -2.303105805000000e+09, 2.442687975000000e+09,
      -1.816357725000000e+09, 8.981988750000000e+08, -2.651825250000000e+08,
      3.535767000000000e+07 };

  static const double fem_coef_gausslob_24[625] =
    { 1.000000000000000e+00, -3.000000000000000e+02, 2.990000000000000e+04,
      -1.480050000000000e+06, 4.351347000000000e+07, -8.412604200000000e+08,
      1.141710570000000e+10, -1.137633032250000e+11, 8.595449577000000e+11,
      -5.042663751840000e+12, 2.337962284944000e+13, -8.678799391080000e+13,
      2.603639817324000e+14, -6.351736697208000e+14, 1.264298066396640e+15,
      -2.054484357894540e+15, 2.719170473683950e+15, -2.914666390092600e+15,
      2.505590405518200e+15, -1.701164012167620e+15, 8.910859111354200e+14,
      -3.471763290138000e+14, 9.468445336740000e+13, -1.612380184155000e+13,
      1.289904147324000e+12, 0.000000000000000e-01, 4.058641271792565e+02,
      -5.527892747506228e+04, 3.080680865091051e+06, -9.608544697986574e+07,
      1.921815551511292e+09, -2.664514371195282e+10, 2.693344195607988e+11,
      -2.055620699316044e+12, 1.214897536784061e+13, -5.664114215003022e+13,
      2.111637034506248e+14, -6.356401824720572e+14, 1.554903769748071e+15,
      -3.101863582054997e+15, 5.049759901096915e+15, -6.693732300831693e+15,
      7.184248666000200e+15, -6.182729556103940e+15, 4.201716500300169e+15,
      -2.202700032171432e+15, 8.588067464630930e+14, -2.343664562972425e+14,
      3.993223187351215e+13, -3.196139551478179e+12, 0.000000000000000e-01,
      -1.624756704002590e+02, 4.076566744849383e+04, -2.856559180759908e+06,
      1.002242332396244e+08, -2.149192199655287e+09, 3.116591524589708e+10,
      -3.248555446646729e+11, 2.534404897909630e+12, -1.522400128496926e+13,
      7.186059820369571e+13, -2.704952845330370e+14, 8.204876425573068e+14,
      -2.019503996546278e+15, 4.049106649016715e+15, -6.619542027745681e+15,
      8.805466617020362e+15, -9.478911036302426e+15, 8.178282706527972e+15,
      -5.570062256804815e+15, 2.925593222099543e+15, -1.142548313369483e+15,
      3.122535907953383e+14, -5.327144417235540e+13, 4.268671053543734e+12,
      0.000000000000000e-01, 9.285790471348058e+01, -2.567292232023452e+04,
      2.172505008713478e+06, -8.632693629032278e+07, 2.009759260312978e+09,
      -3.083881003217180e+10, 3.346965047874316e+11, -2.690206053619648e+12,
      1.652940573703274e+13, -7.940281485029880e+13, 3.030598877175188e+14,
      -9.295754861643471e+14, 2.308921995608790e+15, -4.664329550291535e+15,
      7.673380080847903e+15, -1.026158278560169e+16, 1.109639715400819e+16,
      -9.611028022799504e+15, 6.567868332535721e+15, -3.459765425094874e+15,
      1.354622808613475e+15, -3.710479852977045e+14, 6.342841599973944e+13,
      -5.091588188794019e+12, 0.000000000000000e-01, -6.190263201810896e+01,
      1.771295817306943e+04, -1.605426884400296e+06, 6.937138005497782e+07,
      -1.732266817595608e+09, 2.807090718452186e+10, -3.177491729312126e+11,
      2.638958096874092e+12, -1.663806369766778e+13, 8.158796865135113e+13,
      -3.166342096198069e+14, 9.845663378570931e+14, -2.473337869447428e+15,
      5.044014753850598e+15, -8.364663482728635e+15, 1.126254226203764e+16,
      -1.225027170970453e+16, 1.066427200479037e+16, -7.319785257851210e+15,
      3.870748078365627e+15, -1.520689469004945e+15, 4.177883147633562e+14,
      -7.160927970988626e+13, 5.762006100161049e+12, 0.000000000000000e-01,
      4.501042175430940e+01, -1.308958048352837e+04, 1.225547369430789e+06,
      -5.535760993480313e+07, 1.449945861634959e+09, -2.454369660090688e+10,
      2.883865544213490e+11, -2.470900824125910e+12, 1.598637745987517e+13,
      -8.009303566237079e+13, 3.164491556576764e+14, -9.988976501079246e+14,
      2.541436559580714e+15, -5.239264041161704e+15, 8.769361705836964e+15,
      -1.190220432935039e+16, 1.303612471279278e+16, -1.141726255498977e+16,
      7.878336148801056e+15, -4.185657107040718e+15, 1.651238357905402e+15,
      -4.553312625304459e+14, 7.830179054235796e+13, -6.319165567954503e+12,
      0.000000000000000e-01, -3.460823180120079e+01, 1.015469610544880e+04,
      -9.679531870145017e+05, 4.485134755229398e+07, -1.210735945225935e+09,
      2.114609948086286e+10, -2.559531795223062e+11, 2.252595670603948e+12,
      -1.492191456512238e+13, 7.630937241452699e+13, -3.068986895462340e+14,
      9.837309848042438e+14, -2.536329824251836e+15, 5.289429958022946e+15,
      -8.942835505684019e+15, 1.224496508118832e+16, -1.351567177210176e+16,
      1.191830727462528e+16, -8.273935552638878e+15, 4.419525275428065e+15,
      -1.751884239411906e+15, 4.851652818736986e+14, -8.375521716296401e+13,
      6.782865257514837e+12, 0.000000000000000e-01, 2.766582877253528e+01,
      -8.161941725332768e+03, 7.865526415735044e+05, -3.702889412503837e+07,
      1.019390720188427e+09, -1.819645578676150e+10, 2.252249000861092e+11,
      -2.025483052842698e+12, 1.369084263890056e+13, -7.131369560391163e+13,
      2.915943262851778e+14, -9.485944639479267e+14, 2.478119277999363e+15,
      -5.228788122037598e+15, 8.932615077176731e+15, -1.234455468758280e+16,
      1.373835480205531e+16, -1.220422101884528e+16, 8.528504524807064e+15,
      -4.582579019928659e+15, 1.826238758482371e+15, -5.082003015125067e+14,
      8.811547249928783e+13, -7.164301017225998e+12, 0.000000000000000e-01,
      -2.275670591599455e+01, 6.737590226239552e+03, -6.539504206517175e+05,
      3.111139106156313e+07, -8.679722967903971e+08, 1.573365420642779e+10,
      -1.979909637705803e+11, 1.810880611538906e+12, -1.244463025448231e+13,
      6.585374856883214e+13, -2.732735801573156e+14, 9.011914571652850e+14,
      -2.383831456114384e+15, 5.087291841992724e+15, -8.780953301699755e+15,
      1.224889816733013e+16, -1.374781660040675e+16, 1.230672077620068e+16,
      -8.660226214361099e+15, 4.682883135541219e+15, -1.876981572450838e+15,
      5.250670186065034e+14, -9.147680701891190e+13, 7.470231264323625e+12,
      0.000000000000000e-01, 1.912965144613660e+01, -5.677631395079384e+03,
      5.537935689545739e+05, -2.653927821530689e+07, 7.474036306614312e+08,
      -1.369940651896189e+10, 1.745319507247928e+11, -1.617301631376405e+12,
      1.126327452432594e+13, -6.039298017983792e+13, 2.538313181524926e+14,
      -8.473116290006231e+14, 2.267097817730376e+15, -4.890112526330036e+15,
      8.524655979372967e+15, -1.200076621013731e+16, 1.358349523272153e+16,
      -1.225446423734989e+16, 8.685297402148473e+15, -4.727405822382261e+15,
      1.906317065479628e+15, -5.362492238636365e+14, 9.390516767804315e+13,
      -7.704880889559378e+12, 0.000000000000000e-01, -1.635497697368700e+01,
      4.862656953497025e+03, -4.759804636482565e+05, 2.293041632316653e+07,
      -6.502015538842764e+08, 1.201606379408697e+10, -1.545199230484324e+11,
      1.446437458063616e+12, -1.018096106218994e+13, 5.518468570532575e+13,
      -2.344620385089091e+14, 7.909885622725649e+14, -2.138165417503776e+15,
      4.657339854616682e+15, -8.194527986057532e+15, 1.163730420250179e+16,
      -1.328057957790062e+16, 1.207345046837354e+16, -8.618482475976474e+15,
      4.722435616707614e+15, -1.916176031917556e+15, 5.421466879431477e+14,
      -9.544980641239802e+13, 7.870911362255844e+12, 0.000000000000000e-01,
      1.417066207624127e+01, -4.218698704765500e+03, 4.140273580736256e+05,
      -2.002373113338682e+07, 5.706909517514257e+08, -1.061235738178459e+10,
      1.374488760853391e+11, -1.296867134188179e+12, 9.206001697332345e+12,
      -5.034424091354864e+13, 2.158419782218546e+14, -7.348173021384713e+14,
      2.004252206567683e+15, -4.404149060382797e+15, 7.815178748758451e+15,
      -1.118956431220418e+16, 1.286957131473114e+16, -1.178684055128811e+16,
      8.473168167228462e+15, -4.673705358375533e+15, 1.908297467260950e+15,
      -5.431049066701974e+14, 9.614927445703371e+13, -7.969947411626695e+12,
      0.000000000000000e-01, -1.240846755882427e+01, 3.697723332529632e+03,
      -3.636177333437864e+05, 1.763791694375029e+07, -5.046596469793725e+08,
      9.429433336134134e+09, -1.228099190218494e+11, 1.166008419408402e+12,
      -8.333618884154624e+12, 4.590449180645647e+13, -1.982963080519099e+14,
      6.803133908337801e+14, -1.870091239145240e+15, 4.141349396659428e+15,
      -7.405302748248103e+15, 1.068239700855010e+16, -1.237594459251991e+16,
      1.141465416121965e+16, -8.261228940134621e+15, 4.586380576952005e+15,
      -1.884249466545215e+15, 5.394272826690079e+14, -9.603440259637641e+13,
      8.002866883031367e+12, 0.000000000000000e-01, 1.095558179466470e+01,
      -3.267249008495488e+03, 3.217786804294041e+05, -1.564425754031200e+07,
      4.489762785029423e+08, -8.420409805207488e+09, 1.101506623685581e+11,
      -1.051033145830932e+12, 7.553210200363392e+12, -4.185258869687206e+13,
      1.819278279112502e+14, -6.282336154227903e+14, 1.738507104235632e+15,
      -3.876120340749998e+15, 6.978305450391071e+15, -1.013471839778258e+16,
      1.182005159615111e+16, -1.097352991130819e+16, 7.992846614334175e+15,
      -4.464988119249731e+15, 1.845417602986293e+15, -5.313770797673897e+14,
      9.512946342200696e+13, -7.969947411626695e+12, 0.000000000000000e-01,
      -9.733404845062563e+00, 2.904495367336991e+03, -2.863957452026712e+05,
      1.394908621565543e+07, -4.012835563442479e+08, 7.548227155070059e+09,
      -9.908687724892865e+10, 9.492474279583278e+11, -6.852122094227041e+12,
      3.815223404048325e+13, -1.667054040084672e+14, 5.788252023371522e+14,
      -1.610924210936436e+15, 3.612762299121459e+15, -6.543084510709907e+15,
      9.560030643675167e+15, -1.121725598566102e+16, 1.047660019645038e+16,
      -7.676343347474552e+15, 4.313320840279778e+15, -1.792974677700824e+15,
      5.191726764406062e+14, -9.345206628174223e+13, 7.870911362255844e+12,
      0.000000000000000e-01, 8.685152146887849e+00, -2.592917301003156e+03,
      2.559159080755008e+05, -1.248235381982623e+07, 3.597715754514247e+08,
      -6.783361410773893e+09, 8.929618961811727e+10, -8.582135820103382e+11,
      6.217423158689400e+12, -3.475607425638306e+13, 1.525197220141970e+14,
      -5.320009405626489e+14, 1.487763351500956e+15, -3.353349463718517e+15,
      6.104800041187937e+15, -8.967037300250809e+15, 1.057820092203265e+16,
      -9.933456336414407e+15, 7.318037585534789e+15, -4.134330534453634e+15,
      1.727837357443638e+15, -5.029774927870322e+14, 9.101197367138191e+13,
      -7.704880889559378e+12, 0.000000000000000e-01, -7.768227503689160e+00,
      2.320048262018063e+03, -2.291579825895192e+05, 1.118998178170061e+07,
      -3.230127412726743e+08, 6.101825984880789e+09, -8.050592964465223e+10,
      7.757517929934399e+11, -5.636578411502579e+12, 3.161187883403309e+13,
      -1.392153217132889e+14, 4.874510273126649e+14, -1.368719368735070e+15,
      3.098228256202772e+15, -5.665515780120550e+15, 8.360206053329256e+15,
      -9.909089467205730e+15, 9.350136076665120e+15, -6.922098442148973e+15,
      3.930003596385763e+15, -1.650608740098542e+15, 4.828842861248500e+14,
      -8.780874332485509e+13, 7.470231264323625e+12, 0.000000000000000e-01,
      6.949251795654158e+00, -2.076080736426886e+03, 2.051850668187582e+05,
      -1.002851556374879e+07, 2.898385274463058e+08, -5.483488755332865e+09,
      7.247948130849744e+10, -6.998845716368053e+11, 5.097508994937879e+12,
      -2.866481138828522e+13, 1.266058930533223e+14, -4.447041797385597e+14,
      1.252927498139101e+15, -2.846337193255717e+15, 5.224630116918579e+15,
      -7.740148279482860e+15, 9.211839907560781e+15, -8.729031202403334e+15,
      6.490342376708594e+15, -3.701195553992629e+15, 1.561498591338377e+15,
      -4.588915147832623e+14, 8.382775191413614e+13, -7.164301017225998e+12,
      0.000000000000000e-01, -6.200545901231066e+00, 1.852852310460863e+03,
      -1.832115058838774e+05, 8.961081566680860e+06, -2.592406841374813e+08,
      4.910586592056635e+09, -6.500190149790611e+10, 6.287466876199138e+11,
      -4.588252565931313e+12, 2.585696625207004e+13, -1.144768233740891e+14,
      4.031460020145847e+14, -1.139023606360627e+15, 2.595327972551887e+15,
      -4.779021130665880e+15, 7.103675353349177e+15, -8.483943776806492e+15,
      8.068562477979859e+15, -6.021870692282251e+15, 3.447372991345801e+15,
      -1.460201300789598e+15, 4.308660981996213e+14, -7.903354901739207e+13,
      6.782865257514837e+12, 0.000000000000000e-01, 5.497265260133587e+00,
      -1.643010914375697e+03, 1.625245542407611e+05, -7.953853255844527e+06,
      2.302798044521235e+08, -4.366227075820252e+09, 5.786337032883471e+10,
      -5.604566478207931e+11, 4.096239643506256e+12, -2.312433390636899e+13,
      1.025754064938261e+14, -3.619933583977279e+14, 1.025085118853682e+15,
      -2.341436140046740e+15, 4.322778692729082e+15, -6.443312152718818e+15,
      7.717747123310279e+15, -7.362354286134902e+15, 5.512353176524103e+15,
      -3.166155510128892e+15, 1.345687520087759e+15, -3.984797768116558e+14,
      7.335818308855012e+13, -6.319165567954503e+12, 0.000000000000000e-01,
      -4.814420396783881e+00, 1.439137261510244e+03, -1.424001050225189e+05,
      6.972107765750786e+06, -2.019777804539574e+08, 3.832494908082456e+09,
      -5.083618287186299e+10, 4.929144471099037e+11, -3.606960360420349e+12,
      2.039001482865880e+13, -9.058350359554217e+13, 3.202053356214614e+14,
      -9.083926524953095e+14, 2.078951013598256e+15, -3.846222877659622e+15,
      5.745792065253601e+15, -6.898564020656367e+15, 6.597335811773795e+15,
      -4.952528004193797e+15, 2.852412393699805e+15, -1.215806035913631e+15,
      3.610885650804218e+14, -6.667886669397892e+13, 5.762006100161049e+12,
      0.000000000000000e-01, 4.122502768575858e+00, -1.232445305915745e+03,
      1.219756720552666e+05, -5.974119340666288e+06, 1.731450552812862e+08,
      -3.287266438655767e+09, 4.363384253154989e+10, -4.234185297825935e+11,
      3.101259925467429e+12, -1.754945237689139e+13, 7.805398522969187e+13,
      -2.762644893141324e+14, 7.848217578377621e+14, -1.798840649081661e+15,
      3.333370625337169e+15, -4.988259107765304e+15, 6.000070847701146e+15,
      -5.749271353120764e+15, 4.324788417805135e+15, -2.496262406568326e+15,
      1.066418114121039e+15, -3.174727574108465e+14, 5.876970053131701e+13,
      -5.091588188794019e+12, 0.000000000000000e-01, -3.378098091794216e+00,
      1.009981094027902e+03, -9.997415292076045e+04, 4.897701324351262e+06,
      -1.419932385393241e+08, 2.696914741487987e+09, -3.581511558046563e+10,
      3.477438352489987e+11, -2.548653261914288e+12, 1.443296944374584e+13,
      -6.424560812216257e+13, 2.275969917931459e+14, -6.472060159591627e+14,
      1.485016620793322e+15, -2.755030677045359e+15, 4.127938042773006e+15,
      -4.971860898078912e+15, 4.770796079822418e+15, -3.594142516031414e+15,
      2.077829100777859e+15, -8.891455208945626e+14, 2.651635856092349e+14,
      -4.917666111269423e+13, 4.268671053543734e+12, 0.000000000000000e-01,
      2.493031698759675e+00, -7.454011644125376e+02, 7.379166798110049e+04,
      -3.615566630393531e+06, 1.048426846839652e+08, -1.991802210178859e+09,
      2.645916950749144e+10, -2.569938254028299e+11, 1.884300408926143e+12,
      -1.067564580288449e+13, 4.754491961430585e+13, -1.685282542848975e+14,
      4.795322158961934e+14, -1.101030336950954e+15, 2.044141709763629e+15,
      -3.065196721720791e+15, 3.694953407319221e+15, -3.548705940334544e+15,
      2.676012339710790e+15, -1.548605987126603e+15, 6.633870802695019e+14,
      -1.980596394144405e+14, 3.677511736196415e+13, -3.196139551478179e+12,
      0.000000000000000e-01, -1.000000000000000e+00, 2.990000000000000e+02,
      -2.960100000000000e+04, 1.450449000000000e+06, -4.206302100000000e+07,
      7.991973990000000e+08, -1.061790830100000e+10, 1.031453949240000e+11,
      -7.563995627760000e+11, 4.286264189064000e+12, -1.909335866037600e+13,
      6.769463525042400e+13, -1.926693464819760e+14, 4.425043232388240e+14,
      -8.217937431578160e+14, 1.232690614736724e+15, -1.486479858947226e+15,
      1.428186531145374e+15, -1.077403874372826e+15, 6.237601377947940e+14,
      -2.673257733406260e+14, 7.985055567317400e+13, -1.483389769422600e+13,
      1.289904147324000e+12 };

  static const double fem_coef_gausslob_32[1089] =
    { 1.000000000000000e+00, -5.280000000000000e+02, 9.275200000000000e+04,
      -8.115800000000000e+06, 4.236447600000000e+08, -1.462986571200000e+10,
      3.573867195360000e+11, -6.471252385884000e+12, 8.987850535950000e+13,
      -9.826716585972000e+14, 8.629643838226320e+15, -6.184578084062196e+16,
      3.663173172867608e+17, -1.811459261308158e+18, 7.539120925634905e+18,
      -2.657540126286304e+19, 7.972620378858912e+19, -2.042658293145551e+20,
      4.479513800757788e+20, -8.416770667739633e+20, 1.354699278902855e+21,
      -1.864910695632502e+21, 2.189242990525111e+21, -2.181310950704368e+21,
      1.832301198591669e+21, -1.285429763935079e+21, 7.434251911077520e+20,
      -3.481117958361696e+20, 1.286127324517868e+20, -3.607069737728273e+19,
      7.214139475456547e+18, -9.163120704712953e+17, 5.553406487704820e+16,
      0.000000000000000e-01, 7.143214682034540e+02, -1.714133648848133e+05,
      1.688198752795928e+07, -9.347155797218437e+08, 3.338933321339879e+10,
      -8.331872976904955e+11, 1.530331866800279e+13, -2.146891160964000e+14,
      2.364531150945150e+15,  -2.087974965454055e+16, 1.502766434956253e+17,
      -8.930913114593062e+17, 4.428285547618542e+18, -1.847053591535166e+19,
      6.522650112096549e+19, -1.959752299775458e+20, 5.027465769775919e+20,
      -1.103711061615762e+21, 2.075747280986809e+21, -3.343657099618051e+21,
      4.606186697816854e+21, -5.410587449851726e+21, 5.393904740095235e+21,
      -4.533054367123241e+21, 3.181470251931304e+21, -1.840698151594781e+21,
      8.622096114618880e+20, -3.186487205259235e+20, 8.939310241964093e+19,
      -1.788315002192243e+19, 2.271972224754086e+18, -1.377242653770284e+17,
      0.000000000000000e-01, -2.859599139140576e+02, 1.263498115660723e+05,
      -1.563762114667972e+07, 9.735262028286150e+08, -3.727077001705718e+10,
      9.724728419064618e+11, -1.841437932011095e+13, 2.640184847320257e+14,
      -2.955001754223333e+15, 2.641501188215526e+16, -1.919333187312935e+17,
      1.149300695265413e+18, -5.733477927220639e+18, 2.403400433666191e+19,
      -8.522440623029794e+19, 2.569470483161848e+20, -6.610937005531425e+20,
      1.454972285683492e+21, -2.742254919882459e+21, 4.425523119288710e+21,
      -6.106476088763322e+21, 7.183124069502222e+21, -7.170000472640595e+21,
      6.032425263760434e+21, -4.238001893187230e+21, 2.454158919196046e+21,
      -1.150483843370953e+21, 4.254938488755735e+20, -1.194452209653931e+20,
      2.390927098916194e+19, -3.039204212011252e+18, 1.843238572103207e+17,
      0.000000000000000e-01, 1.634368826450262e+02, -7.956978254181237e+04,
      1.188506225092293e+07, -8.373897346830345e+08, 3.478333876388285e+10,
      -9.598393772867426e+11, 1.891593013667364e+13, -2.793128536050818e+14,
      3.196655214719607e+15, -2.907291848273606e+16, 2.141468780998689e+17,
      -1.296440193900301e+18, 6.525499955570070e+18, -2.755634128784674e+19,
      9.831738858873329e+19, -2.979627701541287e+20, 7.700118175018154e+20,
      -1.701110443974214e+21, 3.216663602425906e+21, -5.205922417479511e+21,
      7.201199888459469e+21, -8.489441777094031e+21, 8.490373918346112e+21,
      -7.155631502801035e+21, 5.034836798969603e+21, -2.919618180375494e+21,
      1.370386674687711e+21, -5.073912120857659e+20, 1.425804015450344e+20,
      -2.856660788223791e+19, 3.634277715764878e+18, -2.205841595819251e+17,
      0.000000000000000e-01, -1.089624682152292e+02, 5.490286557349962e+04,
      -8.781653856365252e+06, 6.724119974793967e+08, -2.993574836604601e+10,
      8.717420300323828e+11, -1.790617757376325e+13, 2.730388108651176e+14,
      -3.204823752640508e+15, 2.974036495407882e+16, -2.226577421064578e+17,
      1.366028725355318e+18, -6.951897867287077e+18, 2.962837694696566e+19,
      -1.065339992825695e+20, 3.250038142137898e+20, -8.446629720680499e+20,
      1.875178351120271e+21, -3.560917379168056e+21, 5.784532874816462e+21,
      -8.027770029015426e+21, 9.491255762425382e+21, -9.516676697643140e+21,
      8.038962462344214e+21, -5.667965587919967e+21, 3.292818343972966e+21,
      -1.548126823651629e+21, 5.740631512413311e+20, -1.615361972833789e+20,
      3.240477969405117e+19, -4.127262310667621e+18, 2.507669351857205e+17,
      0.000000000000000e-01, 7.924220268285228e+01, -4.057928795465258e+04,
      6.704332297502558e+06, -5.364604956918865e+08, 2.503646198522573e+10,
      -7.610195551213778e+11, 1.621371476972958e+13, -2.548664517300070e+14,
      3.067722734616486e+15, -2.906734241270257e+16, 2.214249975700081e+17,
      -1.378338812356282e+18, 7.101001568235638e+18, -3.058038376939241e+19,
      1.109399051188740e+20, -3.410471802537065e+20, 8.922579803599067e+20,
      -1.992320553154499e+21, 3.802564715982721e+21, -6.204660474711801e+21,
      8.644825045503473e+21, -1.025665248235139e+22, 1.031630159337718e+22,
      -8.738843624083011e+21, 6.176944575139345e+21, -3.596664164242111e+21,
      1.694457654020145e+21, -6.294971162057861e+20, 1.774358410868178e+20,
      -3.564953335849152e+19, 4.546981308698057e+18, -2.766285114923478e+17,
      0.000000000000000e-01, -6.094810716099949e+01, 3.149084615481936e+04,
      -5.296674502372112e+06, 4.346997745238327e+08, -2.090081537881765e+10,
      6.551264846374850e+11, -1.436792716105805e+13, 2.318076448212311e+14,
      -2.854539793592080e+15, 2.758693070051896e+16, -2.137570364709653e+17,
      1.350278421513405e+18, -7.045142142783022e+18, 3.067459655658656e+19,
      -1.123483766643411e+20, 3.482651662900422e+20, -9.178174833360337e+20,
      2.062604456625362e+21, -3.959135109240898e+21, 6.492786887393485e+21,
      -9.086987198956428e+21, 1.082464243206250e+22, -1.092690369776145e+22,
      9.286162259274268e+21, -6.583075803335110e+21, 3.843334757170004e+21,
      -1.815042549975247e+21, 6.757786079688690e+20, -1.908640091400089e+20,
      3.841802724253007e+19, -4.908370532055551e+18, 2.990786503802649e+17,
      0.000000000000000e-01, 4.874762271398016e+01, -2.532459924379460e+04,
      4.306289111489640e+06, -3.590409832532830e+08, 1.760136771205027e+10,
      -5.636351004936871e+11, 1.263327395525212e+13, -2.081295681322519e+14,
      2.613155513828582e+15, -2.570230208401814e+16, 2.023153786109967e+17,
      -1.296022489490302e+18, 6.846470251956364e+18, -3.013872322115386e+19,
      1.114644422978128e+20, -3.485183237291328e+20, 9.255531281005032e+20,
      -2.094244828319674e+21, 4.044473148145946e+21, -6.669094801072411e+21,
      9.379689895649970e+21, -1.122286017421572e+22, 1.137425276024759e+22,
      -9.701397530693913e+21, 6.900090712824613e+21, -4.040491989417283e+21,
      1.913372195656766e+21, -7.141717651945982e+20, 2.021704584417841e+20,
      -4.077964478229480e+19, 5.220210907153068e+18, -3.186495777814819e+17,
      0.000000000000000e-01, -4.013085745911482e+01, 2.092265556924549e+04,
      -3.583307397906160e+06, 3.019036845311046e+08, -1.499682567334736e+10,
      4.875419883971985e+11, -1.110534210962378e+13, 1.859662144990967e+14,
      -2.372232833495577e+15, 2.368570555021916e+16, -1.890606460337203e+17,
      1.226710982466906e+18, -6.556236864370271e+18, 2.916718348444145e+19,
      -1.089043462231537e+20, 3.434548724554294e+20, -9.192120758302752e+20,
      2.094521354663323e+21, -4.070706954543361e+21, 6.750946202167051e+21,
      -9.544297656724117e+21, 1.147387384551836e+22, -1.167874642179135e+22,
      1.000023491239336e+22, -7.138163819152158e+21, 4.193633752898508e+21,
      -1.991877420085289e+21, 7.455334138647318e+20, -2.115867266165027e+20,
      4.277941431548355e+19, -5.488110031116298e+18, 3.356769581362759e+17,
      0.000000000000000e-01, 3.377658303742900e+01, -1.765321539233928e+04,
      3.038340440514934e+06, -2.578584606807322e+08, 1.292884612296167e+10,
      -4.249332478575872e+11, 9.796453340445514e+12, -1.661322016030880e+14,
      2.146412285165976e+15, -2.170063067679739e+16, 1.753071524663780e+17,
      -1.150445213494342e+18, 6.214123504683469e+18, -2.791805131931260e+19,
      1.051885109311114e+20, -3.345072958360405e+20, 9.021185531229185e+20,
      -2.069976464960887e+21, 4.048800325514122e+21, -6.754020775002269e+21,
      9.599959543792231e+21, -1.159763408249418e+22, 1.185806715330365e+22,
      -1.019594067425241e+22, 7.305655578610319e+21, -4.307135697275206e+21,
      2.052428738060058e+21, -7.705007825196310e+20, 2.192794049399991e+20,
      -4.444874922276286e+19, 5.715873383290923e+18, -3.503832522669480e+17,
      0.000000000000000e-01, -2.892964123166297e+01, 1.514678464159201e+04,
      -2.616230196259168e+06, 2.232056370284014e+08, -1.126780271399590e+10,
      3.733563821201946e+11, -8.686293054983950e+12, 1.487584698214981e+14,
      -1.941627928264148e+15, 1.983312707085937e+16, -1.618550810194144e+17,
      1.072675099399285e+18, -5.848903282733471e+18, 2.651290247911275e+19,
      -1.007365700808825e+20, 3.228755039109242e+20, -8.771430385090572e+20,
      2.026394505724555e+21, -3.988616003559215e+21, 6.692583651440911e+21,
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      6.746687832309763e+20, -1.406360250100074e+21, 2.497787658823857e+21,
      -3.775905442111079e+21, 4.846020652107188e+21, -5.257496778824818e+21,
      4.790865278530114e+21, -3.633565158982738e+21, 2.264711978580421e+21,
      -1.139484606704540e+21, 4.511274997631573e+20, -1.352342175988863e+20,
      2.884004791547230e+19, -3.897279615275436e+18, 2.507669351857205e+17,
      0.000000000000000e-01, 4.070989770987400e+00, -2.145310206509185e+03,
      3.753936962817004e+05, -3.265459454227024e+07, 1.691185508550510e+09,
      -5.782472303217436e+10, 1.395652621524724e+12, -2.491398584416296e+13,
      3.403599167085817e+14, -3.651608452505086e+15, 3.138862675548854e+16,
      -2.196030665561983e+17, 1.266200972268635e+18, -6.076691812326445e+18,
      2.446363025550984e+19, -8.311520079137181e+19, 2.393790730533568e+20,
      -5.862224225217512e+20, 1.222781062900224e+21, -2.173219858365351e+21,
      3.287615107047680e+21, -4.222527251629046e+21, 4.584681175928391e+21,
      -4.181215238485294e+21, 3.173915831158933e+21, -1.979997675938663e+21,
      9.971596317435381e+20, -3.951620422561584e+20, 1.185761286177830e+20,
      -2.531374184616153e+19, 3.424415390856724e+18, -2.205841595819251e+17,
      0.000000000000000e-01, -3.358090451409268e+00, 1.769674233094550e+03,
      -3.096791496404924e+05, 2.694027470522958e+07, -1.395380783042074e+09,
      4.771664530498290e+10, -1.151859937920620e+12, 2.056566436202355e+13,
      -2.810129791321012e+14, 3.015587336970416e+15, -2.592812452685370e+16,
      1.814511218788989e+17, -1.046544742872117e+18, 5.024209499496014e+18,
      -2.023384009065594e+19, 6.877115067771995e+19, -1.981490581745830e+20,
      4.854671646250063e+20, -1.013093139325803e+21, 1.801437605201838e+21,
      -2.726610427990654e+21, 3.503909183048419e+21, -3.806619619529808e+21,
      3.473717880327268e+21, -2.638522642188920e+21, 1.647082019674793e+21,
      -8.300649678444784e+20, 3.291782934571713e+20, -9.884928188742345e+19,
      2.111857359313218e+19, -2.859159218719010e+18, 1.843238572103207e+17,
      0.000000000000000e-01, 2.488636218117664e+00, -1.311502616747683e+03,
      2.295099147207366e+05, -1.996696431094197e+07, 1.034261165808075e+09,
      -3.537054923192207e+10, 8.539117568460436e+11, -1.524770635014292e+13,
      2.083740755848528e+14, -2.236412246676083e+15, 1.923184048547382e+16,
      -1.346128075294067e+17, 7.765487558375990e+17, -3.728808938617731e+18,
      1.502032959138465e+19, -5.106384693104742e+19, 1.471677691807719e+20,
      -3.606628515829967e+20, 7.528686576360846e+20, -1.339136443296313e+21,
      2.027551807534738e+21, -2.606468672347013e+21, 2.832680005398635e+21,
      -2.585940609411422e+21, 1.964981315222245e+21, -1.227139920687813e+21,
      6.186996825719858e+20, -2.454684425809199e+20, 7.374666999744300e+19,
      -1.576324668155186e+19, 2.135204267310823e+18, -1.377242653770284e+17,
      0.000000000000000e-01, -1.000000000000000e+00, 5.270000000000000e+02,
      -9.222500000000000e+04, 8.023575000000000e+06, -4.156211850000000e+08,
      1.421424452700000e+10, -3.431724750090000e+11, 6.128079910875000e+12,
      -8.375042544862500e+13, 8.989212331485750e+14, -7.730722605077745e+15,
      5.411505823554421e+16, -3.122022590512166e+17, 1.499257002256941e+18,
      -6.039863923377964e+18, 2.053553733948508e+19, -5.919066644910405e+19,
      1.450751628654511e+20, -3.028762172103277e+20, 5.388008495636356e+20,
      -8.158984293392197e+20, 1.049012266293282e+21, -1.140230724231829e+21,
      1.041080226472539e+21, -7.912209721191298e+20, 4.942087918159488e+20,
      -2.492163992918032e+20, 9.889539654436636e+19, -2.971733590742043e+19,
      6.353361469862300e+18, -8.607780055942471e+17, 5.553406487704820e+16 };

  static const double *fem_coeff_gausslob[] =
    { 0, fem_coef_gausslob_1, fem_coef_gausslob_2, fem_coef_gausslob_3,
      fem_coef_gausslob_4, fem_coef_gausslob_5, fem_coef_gausslob_6, fem_coef_gausslob_7, fem_coef_gausslob_8,
      fem_coef_gausslob_9, fem_coef_gausslob_10, fem_coef_gausslob_11, fem_coef_gausslob_12, fem_coef_gausslob_13,
      fem_coef_gausslob_14, 0, fem_coef_gausslob_16, 0, 0, 0, 0, 0, 0, 0, fem_coef_gausslob_24, 0, 0, 0, 0, 0, 0, 0,
      fem_coef_gausslob_32, };

  const unsigned fem_coeff_gausslob_max_k = 33;


  class PK_GL_fem_ : public fem<base_poly> {
  public :
    PK_GL_fem_(unsigned k);
  };

  PK_GL_fem_::PK_GL_fem_(unsigned k) {
    cvr = bgeot::simplex_of_reference(1);
    dim_ = cvr->structure()->dim();
    is_standard_fem = is_equiv = is_pol = is_lag = true;
    es_degree = short_type(k);
    GMM_ASSERT1(k < fem_coeff_gausslob_max_k && fem_coeff_gausslob[k],
                "try another degree");
    init_cvs_node();
    std::stringstream sstr; sstr << "IM_GAUSSLOBATTO1D(" << k*2-1 << ")";
    pintegration_method gl_im = int_method_descriptor(sstr.str());
    std::vector<base_node> points(k+1);
    for (size_type i = 0; i < k+1; ++i) {
      points[i] = gl_im->approx_method()->point(i);
    }
    std::sort(points.begin(),points.end());
    for (size_type i = 0; i < k+1; ++i) {
      // cout << points[i][0] << endl;
      add_node(lagrange_dof(1), points[i]);
    }
    base_.resize(k+1);
    const double *coefs = fem_coeff_gausslob[k];
    for (size_type r = 0; r < k+1; r++) {
      base_[r] = base_poly(1,short_type(k));
      std::copy(coefs, coefs+k+1, base_[r].begin());
      coefs += k+1;
    }
  }

  static pfem PK_GL_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int k = int(::floor(params[0].num() + 0.01));
    pfem p = std::make_shared<PK_GL_fem_>(k);
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*        Hermite element on the segment                                    */
  /* ******************************************************************** */

  struct hermite_segment__ : public fem<base_poly> {
    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    hermite_segment__();
  };

  void hermite_segment__::mat_trans(base_matrix &M,
                                    const base_matrix &G,
                                    bgeot::pgeometric_trans pgt) const {
    DEFINE_STATIC_THREAD_LOCAL(bgeot::pgeotrans_precomp, pgp);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_stored, 0);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, K, 1, 1);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_vector,r, 1);
    dim_type N = dim_type(G.nrows());

    if (pgt != pgt_stored) {
      gmm::resize(r, N);
      for (size_type i = 0; i < N; ++i) r[i] = ::exp(double(i));
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0);
      gmm::resize(K, N, 1);
    }
    gmm::copy(gmm::identity_matrix(), M);
    // gradient at point 0
    gmm::mult(G, pgp->grad(1), K);
    if (N == 1) M(1, 1) = K(0,0);
    else  M(1, 1) = gmm::mat_euclidean_norm(K)
      * gmm::sgn(gmm::vect_sp(gmm::mat_col(K, 0), r));
    // gradient at point 1
    if (!(pgt->is_linear())) gmm::mult(G, pgp->grad(3), K);
    if (N == 1) M(3, 3) = K(0,0);
    else M(3, 3) = gmm::mat_euclidean_norm(K)
      * gmm::sgn(gmm::vect_sp(gmm::mat_col(K, 0), r));
  }

  // Hermite element on the segment. when the real element lies in
  // a 2 or 3 dimensional domain, the element should still work if
  // the tangent coincides.
  hermite_segment__::hermite_segment__() {
    base_node pt(1);
    cvr = bgeot::simplex_of_reference(1);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 3;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    base_.resize(4);

    pt[0] = 0.0; add_node(lagrange_dof(1), pt);
    read_poly(base_[0], 1, "(1 - x)^2*(2*x + 1)");

    pt[0] = 0.0; add_node(derivative_dof(1, 0), pt);
    read_poly(base_[1], 1, "x*(x - 1)*(x - 1)");

    pt[0] = 1.0; add_node(lagrange_dof(1), pt);
    read_poly(base_[2], 1, "x*x*(3  - 2*x)");

    pt[0] = 1.0; add_node(derivative_dof(1, 0), pt);
    read_poly(base_[3], 1, "x*x*(x - 1)");
  }

  /* ******************************************************************** */
  /*        Hermite element on the triangle                                   */
  /* ******************************************************************** */

  struct hermite_triangle__ : public fem<base_poly> {
    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    hermite_triangle__();
  };

  void hermite_triangle__::mat_trans(base_matrix &M,
                                    const base_matrix &G,
                                    bgeot::pgeometric_trans pgt) const {

    DEFINE_STATIC_THREAD_LOCAL(bgeot::pgeotrans_precomp, pgp);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_stored, 0);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, K, 2, 2);
    dim_type N = dim_type(G.nrows());

    GMM_ASSERT1(N == 2, "Sorry, this version of hermite "
                "element works only in dimension two.")
    if (pgt != pgt_stored)
      { pgt_stored = pgt; pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0); }
    gmm::copy(gmm::identity_matrix(), M);

    gmm::mult(G, pgp->grad(0), K);
    for (size_type i = 0; i < 3; ++i) {
      if (i && !(pgt->is_linear())) gmm::mult(G, pgp->grad(i*3), K);
      gmm::copy(K, gmm::sub_matrix(M, gmm::sub_interval(1+3*i, 2)));
    }
  }

  hermite_triangle__::hermite_triangle__() {
    cvr = bgeot::simplex_of_reference(2);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 3;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    base_.resize(10);

    add_node(lagrange_dof(2), base_node(0.0, 0.0));
    read_poly(base_[0], 2, "(1 - x - y)*(1 + x + y - 2*x*x - 11*x*y - 2*y*y)");

    add_node(derivative_dof(2, 0), base_node(0.0, 0.0));
    read_poly(base_[1], 2, "x*(1 - x - y)*(1 - x - 2*y)");

    add_node(derivative_dof(2, 1), base_node(0.0, 0.0));
    read_poly(base_[2], 2, "y*(1 - x - y)*(1 - 2*x - y)");

    add_node(lagrange_dof(2), base_node(1.0, 0.0));
    read_poly(base_[3], 2, "-2*x*x*x + 7*x*x*y + 7*x*y*y + 3*x*x - 7*x*y");

    add_node(derivative_dof(2, 0), base_node(1.0, 0.0));
    read_poly(base_[4], 2, "x*x*x - 2*x*x*y - 2*x*y*y - x*x + 2*x*y");

    add_node(derivative_dof(2, 1), base_node(1.0, 0.0));
    read_poly(base_[5], 2, "x*y*(2*x + y - 1)");

    add_node(lagrange_dof(2), base_node(0.0, 1.0));
    read_poly(base_[6], 2, "7*x*x*y + 7*x*y*y - 2*y*y*y + 3*y*y - 7*x*y");

    add_node(derivative_dof(2, 0), base_node(0.0, 1.0));
    read_poly(base_[7], 2, "x*y*(x + 2*y - 1)");

    add_node(derivative_dof(2, 1), base_node(0.0, 1.0));
    read_poly(base_[8], 2, "y*y*y - 2*y*y*x - 2*y*x*x - y*y + 2*x*y");

    add_node(lagrange_dof(2), base_node(1.0/3.0, 1.0/3.0));
    read_poly(base_[9], 2, "27*x*y*(1 - x - y)");

 }

  /* ******************************************************************** */
  /*        Hermite element on the tetrahedron                                */
  /* ******************************************************************** */

  struct hermite_tetrahedron__ : public fem<base_poly> {
    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    hermite_tetrahedron__();
  };

  void hermite_tetrahedron__::mat_trans(base_matrix &M,
                                    const base_matrix &G,
                                    bgeot::pgeometric_trans pgt) const {
    DEFINE_STATIC_THREAD_LOCAL(bgeot::pgeotrans_precomp, pgp);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_stored, 0);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, K, 3, 3);
    dim_type N = dim_type(G.nrows());
    GMM_ASSERT1(N == 3, "Sorry, this version of hermite "
                "element works only on dimension three.")
    if (pgt != pgt_stored)
      { pgt_stored = pgt; pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0); }
    gmm::copy(gmm::identity_matrix(), M);

    gmm::mult(G, pgp->grad(0), K);
    for (size_type k = 0; k < 4; ++k) {
      if (k && !(pgt->is_linear())) gmm::mult(G, pgp->grad(k*4), K);
      gmm::copy(K, gmm::sub_matrix(M, gmm::sub_interval(1+4*k, 3)));
    }
  }

  hermite_tetrahedron__::hermite_tetrahedron__() {
    cvr = bgeot::simplex_of_reference(3);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 3;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    base_.resize(20);
    std::stringstream s
      ( "1 - 3*x*x - 13*x*y - 13*x*z - 3*y*y - 13*y*z - 3*z*z + 2*x*x*x"
        "+ 13*x*x*y + 13*x*x*z + 13*x*y*y + 33*x*y*z + 13*x*z*z + 2*y*y*y"
        "+ 13*y*y*z + 13*y*z*z + 2*z*z*z;"
        "x - 2*x*x - 3*x*y - 3*x*z + x*x*x + 3*x*x*y + 3*x*x*z + 2*x*y*y"
        "+ 4*x*y*z + 2*x*z*z;"
        "y - 3*x*y - 2*y*y - 3*y*z + 2*x*x*y + 3*x*y*y + 4*x*y*z"
        "+ y*y*y + 3*y*y*z + 2*y*z*z;"
        "z - 3*x*z - 3*y*z - 2*z*z + 2*x*x*z + 4*x*y*z + 3*x*z*z"
        "+ 2*y*y*z + 3*y*z*z + z*z*z;"
        "3*x*x - 7*x*y - 7*x*z - 2*x*x*x + 7*x*x*y + 7*x*x*z + 7*x*y*y"
        "+ 7*x*y*z + 7*x*z*z;"
        "-x*x + 2*x*y + 2*x*z + x*x*x - 2*x*x*y - 2*x*x*z - 2*x*y*y"
        "- 2*x*y*z - 2*x*z*z;"
        "-x*y + 2*x*x*y + x*y*y;"
        "-x*z + 2*x*x*z + x*z*z;"
        "-7*x*y + 3*y*y - 7*y*z + 7*x*x*y + 7*x*y*y + 7*x*y*z - 2*y*y*y"
        "+ 7*y*y*z + 7*y*z*z;"
        "-x*y + x*x*y + 2*x*y*y;"
        "2*x*y - y*y + 2*y*z - 2*x*x*y - 2*x*y*y - 2*x*y*z + y*y*y"
        "- 2*y*y*z - 2*y*z*z;"
        "-y*z + 2*y*y*z + y*z*z;"
        "-7*x*z - 7*y*z + 3*z*z + 7*x*x*z + 7*x*y*z + 7*x*z*z + 7*y*y*z"
        "+ 7*y*z*z - 2*z*z*z;"
        "-x*z + x*x*z + 2*x*z*z;"
        "-y*z + y*y*z + 2*y*z*z;"
        "2*x*z + 2*y*z - z*z - 2*x*x*z - 2*x*y*z - 2*x*z*z - 2*y*y*z"
        "- 2*y*z*z + z*z*z;"
        "27*x*y*z;"
        "27*y*z - 27*x*y*z - 27*y*y*z - 27*y*z*z;"
        "27*x*z - 27*x*x*z - 27*x*y*z - 27*x*z*z;"
        "27*x*y - 27*x*x*y - 27*x*y*y - 27*x*y*z;");

    base_node pt(3);
    for (unsigned k = 0; k < 5; ++k) {
      for (unsigned i = 0; i < 4; ++i) {
        base_[k*4+i] = read_base_poly(3, s);
        pt[0] = pt[1] = pt[2] = ((k == 4) ? 1.0/3.0 : 0.0);
        if (k == 4 && i) pt[i-1] = 0.0;
        if (k < 4 && k) pt[k-1] = 1.0;
        if (k == 4 || i == 0)  add_node(lagrange_dof(3), pt);
        else add_node(derivative_dof(3, dim_type(i-1)), pt);
      }
    }
  }

  static pfem Hermite_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 1, "Bad number of parameters : "
                << params.size() << " should be 1.");
    GMM_ASSERT1(params[0].type() == 0, "Bad type of parameters");
    int d = int(::floor(params[0].num() + 0.01));
    pfem p;
    switch(d) {
    case 1 : p = std::make_shared<hermite_segment__>(); break;
    case 2 : p = std::make_shared<hermite_triangle__>(); break;
    case 3 : p = std::make_shared<hermite_tetrahedron__>(); break;
    default : GMM_ASSERT1(false, "Sorry, Hermite element in dimension "
                          << d << " not available");
    }
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return pfem(p);
  }

  /* ******************************************************************** */
  /*    Argyris element on the triangle                                   */
  /* ******************************************************************** */

  struct argyris_triangle__ : public fem<base_poly> {
    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    argyris_triangle__();
  };

  void argyris_triangle__::mat_trans(base_matrix &M,
                                    const base_matrix &G,
                                    bgeot::pgeometric_trans pgt) const {

    DEFINE_STATIC_THREAD_LOCAL(bgeot::pgeotrans_precomp, pgp);
    DEFINE_STATIC_THREAD_LOCAL(pfem_precomp, pfp);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_stored, 0);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, K, 2, 2);
    dim_type N = dim_type(G.nrows());
    GMM_ASSERT1(N == 2, "Sorry, this version of argyris "
                "element works only on dimension two.")
    if (pgt != pgt_stored) {
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0);
      pfp = fem_precomp(std::make_shared<argyris_triangle__>(), node_tab(0), 0);
    }
    gmm::copy(gmm::identity_matrix(), M);

    gmm::mult(G, pgp->grad(0), K); // gradient at point (0, 0)
    for (unsigned k = 0; k < 3; ++k) {
      if (k && !(pgt->is_linear())) gmm::mult(G, pgp->grad(6*k), K);
      M(1+6*k, 1+6*k) = K(0,0); M(1+6*k, 2+6*k) = K(0,1);
      M(2+6*k, 1+6*k) = K(1,0); M(2+6*k, 2+6*k) = K(1,1);
      if (!(pgt->is_linear())) {
        base_matrix XX[2], H(2,4), B(2,2), X(2,2);
        XX[0] = XX[1] = base_matrix(2,2);
        gmm::copy(gmm::transposed(K), B); gmm::lu_inverse(B);
        gmm::mult(G, pgp->hessian(6*k), H);
        for (unsigned j = 0; j < 2; ++j) {
          XX[j](0,0) = B(0, j)*H(0, 0) + B(1, j)*H(1, 0);
          XX[j](0,1) = XX[j](1,0) = B(0, j)*H(0, 1) + B(1, j)*H(1, 1);
          XX[j](1,1) = B(0, j)*H(0, 3) + B(1, j)*H(1, 3);
        }
        for (unsigned j = 0; j < 2; ++j) {
          gmm::copy(gmm::scaled(XX[0], K(j,0)), X);
          gmm::add(gmm::scaled(XX[1], K(j,1)), X);
          M(1+j+6*k, 3+6*k) = X(0,0); M(1+j+6*k, 4+6*k) = X(1, 0);
          M(1+j+6*k, 5+6*k) = X(1, 1);
        }
      }
      scalar_type a = K(0,0), b = K(0,1), c = K(1,0), d = K(1,1);
      M(3+6*k, 3+6*k) = a*a;     M(3+6*k, 4+6*k) = a*b;       M(3+6*k, 5+6*k) = b*b;
      M(4+6*k, 3+6*k) = 2.0*a*c; M(4+6*k, 4+6*k) = b*c + a*d; M(4+6*k, 5+6*k) = 2.0*b*d;
      M(5+6*k, 3+6*k) = c*c;     M(5+6*k, 4+6*k) = c*d;       M(5+6*k, 5+6*k) = d*d;
    }

    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, W, 3, 21);
    base_small_vector norient(M_PI, M_PI * M_PI);
    if (pgt->is_linear()) gmm::lu_inverse(K);
    for (unsigned i = 18; i < 21; ++i) {
      if (!(pgt->is_linear()))
        { gmm::mult(G, pgp->grad(i), K); gmm::lu_inverse(K); }
      bgeot::base_small_vector n(2), v(2);
      gmm::mult(gmm::transposed(K), cvr->normals()[i-18], n);
      n /= gmm::vect_norm2(n);

      scalar_type ps = gmm::vect_sp(n, norient);
      if (ps < 0) n *= scalar_type(-1);
      if (gmm::abs(ps) < 1E-8)
        GMM_WARNING2("Argyris : The normal orientation may be not correct");
      gmm::mult(K, n, v);
      const bgeot::base_tensor &t = pfp->grad(i);
      for (unsigned j = 0; j < 21; ++j)
        W(i-18, j) = t(j, 0, 0) * v[0] + t(j, 0, 1) * v[1];
    }
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix,A,3,3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(bgeot::base_vector, w, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(bgeot::base_vector, coeff, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(gmm::sub_interval, SUBI, 18,3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(gmm::sub_interval, SUBJ, 0,3);
    gmm::copy(gmm::sub_matrix(W, SUBJ, SUBI), A);
    gmm::lu_inverse(A);
    gmm::copy(gmm::transposed(A), gmm::sub_matrix(M, SUBI));

    for (unsigned j = 0; j < 18; ++j) {
      gmm::mult(W, gmm::mat_row(M, j), w);
      gmm::mult(A, gmm::scaled(w, -1.0), coeff);
      gmm::copy(coeff, gmm::sub_vector(gmm::mat_row(M, j), SUBI));
    }
  }

  argyris_triangle__::argyris_triangle__() {
    cvr = bgeot::simplex_of_reference(2);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 5;
    is_pol = true;
    is_lag = false;
    is_standard_fem = is_equiv = false;
    base_.resize(21);

    std::stringstream s
      ("1 - 10*x^3 - 10*y^3 + 15*x^4 - 30*x*x*y*y"
       "+ 15*y*y*y*y - 6*x^5 + 30*x*x*x*y*y + 30*x*x*y*y*y - 6*y^5;"
       "x - 6*x*x*x - 11*x*y*y + 8*x*x*x*x + 10*x*x*y*y"
       "+ 18*x*y*y*y - 3*x*x*x*x*x + x*x*x*y*y - 10*x*x*y*y*y - 8*x*y*y*y*y;"
       "y - 11*x*x*y - 6*y*y*y + 18*x*x*x*y + 10*x*x*y*y"
       "+ 8*y*y*y*y - 8*x*x*x*x*y - 10*x*x*x*y*y + x*x*y*y*y - 3*y*y*y*y*y;"
       "0.5*x*x - 1.5*x*x*x + 1.5*x*x*x*x - 1.5*x*x*y*y"
       "- 0.5*x*x*x*x*x + 1.5*x*x*x*y*y + x*x*y*y*y;"
       "x*y - 4*x*x*y - 4*x*y*y + 5*x*x*x*y + 10*x*x*y*y"
       "+ 5*x*y*y*y - 2*x*x*x*x*y - 6*x*x*x*y*y - 6*x*x*y*y*y - 2*x*y*y*y*y;"
       "0.5*y*y - 1.5*y*y*y - 1.5*x*x*y*y + 1.5*y*y*y*y + x*x*x*y*y"
       "+ 1.5*x*x*y*y*y - 0.5*y*y*y*y*y;"
       "10*x^3 - 15*x^4 + 15*x*x*y*y + 6*x^5 - 15*x*x*x*y*y - 15*x*x*y*y*y;"
       "-4*x*x*x + 7*x*x*x*x - 3.5*x*x*y*y - 3*x*x*x*x*x + 3.5*x*x*x*y*y"
       "+ 3.5*x*x*y*y*y;"
       "-5*x*x*y + 14*x*x*x*y + 18.5*x*x*y*y - 8*x*x*x*x*y"
       "- 18.5*x*x*x*y*y - 13.5*x*x*y*y*y;"
       "0.5*x*x*x - x*x*x*x + 0.25*x*x*y*y + 0.5*x*x*x*x*x"
       "- 0.25*x*x*x*y*y - 0.25*x*x*y*y*y;"
       "x*x*y - 3*x*x*x*y - 3.5*x*x*y*y + 2*x*x*x*x*y + 3.5*x*x*x*y*y"
       "+ 2.5*x*x*y*y*y;"
       "1.25*x*x*y*y - 0.75*x*x*x*y*y - 1.25*x*x*y*y*y;"
       "10*y*y*y + 15*x*x*y*y - 15*y^4 - 15*x*x*x*y*y - 15*x*x*y*y*y + 6*y^5;"
       "-5*x*y*y + 18.5*x*x*y*y + 14*x*y*y*y - 13.5*x*x*x*y*y"
       "- 18.5*x*x*y*y*y - 8*x*y*y*y*y;"
       "-4*y*y*y - 3.5*x*x*y*y + 7*y*y*y*y + 3.5*x*x*x*y*y"
       "+ 3.5*x*x*y*y*y - 3*y*y*y*y*y;"
       "1.25*x*x*y*y - 1.25*x*x*x*y*y - 0.75*x*x*y*y*y;"
       "x*y*y - 3.5*x*x*y*y - 3*x*y*y*y + 2.5*x*x*x*y*y + 3.5*x*x*y*y*y"
       "+ 2*x*y*y*y*y;"
       "0.5*y*y*y + 0.25*x*x*y*y - y*y*y*y - 0.25*x*x*x*y*y"
       "- 0.25*x*x*y*y*y + 0.5*y*y*y*y*y;"
       "sqrt(2) * (-8*x*x*y*y + 8*x*x*x*y*y + 8*x*x*y*y*y);"
       "-16*x*y*y + 32*x*x*y*y + 32*x*y*y*y - 16*x*x*x*y*y"
       "- 32*x*x*y*y*y - 16*x*y*y*y*y;"
       "-16*x*x*y + 32*x*x*x*y + 32*x*x*y*y - 16*x*x*x*x*y"
       "- 32*x*x*x*y*y - 16*x*x*y*y*y;");

    base_node pt(2);
    for (unsigned k = 0; k < 7; ++k) {
      for (unsigned i = 0; i < 3; ++i) {
        base_[k*3+i] = read_base_poly(2, s);
        if (k == 6) {
          pt[0] = pt[1] = 0.5; if (i) pt[i-1] = 0.0;
          add_node(normal_derivative_dof(2), pt);
        }
        else {
          pt[0] = pt[1] = 0.0; if (k/2) pt[k/2-1] = 1.0;
          if (k & 1)
            add_node(second_derivative_dof(2, dim_type((i) ? 1:0),
                                           dim_type((i == 2) ? 1:0)), pt);
          else {
            if (i) add_node(derivative_dof(2, dim_type(i-1)), pt);
            else add_node(lagrange_dof(2), pt);
          }
        }
      }
    }
  }

  static pfem triangle_Argyris_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = std::make_shared<argyris_triangle__>();
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*    Morley element on the triangle                                    */
  /* ******************************************************************** */

  struct morley_triangle__ : public fem<base_poly> {
    virtual void mat_trans(base_matrix &M, const base_matrix &G,
                           bgeot::pgeometric_trans pgt) const;
    morley_triangle__();
  };

  void morley_triangle__::mat_trans(base_matrix &M,
                                    const base_matrix &G,
                                    bgeot::pgeometric_trans pgt) const {

    DEFINE_STATIC_THREAD_LOCAL(bgeot::pgeotrans_precomp, pgp);
    DEFINE_STATIC_THREAD_LOCAL(pfem_precomp, pfp);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_stored, 0);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, K, 2, 2);
    dim_type N = dim_type(G.nrows());
    GMM_ASSERT1(N == 2, "Sorry, this version of morley "
                "element works only on dimension two.")

    if (pgt != pgt_stored) {
      pgt_stored = pgt;
      pgp = bgeot::geotrans_precomp(pgt, node_tab(0), 0);
      pfp = fem_precomp(std::make_shared<morley_triangle__>(), node_tab(0), 0);
    }
    gmm::copy(gmm::identity_matrix(), M);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, W, 3, 6);
    base_small_vector norient(M_PI, M_PI * M_PI);
    if (pgt->is_linear())
      { gmm::mult(G, pgp->grad(0), K); gmm::lu_inverse(K); }
    for (unsigned i = 3; i < 6; ++i) {
      if (!(pgt->is_linear()))
        { gmm::mult(G, pgp->grad(i), K); gmm::lu_inverse(K); }
      bgeot::base_small_vector n(2), v(2);
      gmm::mult(gmm::transposed(K), cvr->normals()[i-3], n);
      n /= gmm::vect_norm2(n);

      scalar_type ps = gmm::vect_sp(n, norient);
      if (ps < 0) n *= scalar_type(-1);
      if (gmm::abs(ps) < 1E-8)
        GMM_WARNING2("Morley : The normal orientation may be not correct");
      gmm::mult(K, n, v);
      const bgeot::base_tensor &t = pfp->grad(i);
      for (unsigned j = 0; j < 6; ++j)
        W(i-3, j) = t(j, 0, 0) * v[0] + t(j, 0, 1) * v[1];
    }
    //    cout << "W = " << W << endl; getchar();

    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_matrix, A, 3, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_vector, w, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(base_vector, coeff, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(gmm::sub_interval, SUBI, 3, 3);
    DEFINE_STATIC_THREAD_LOCAL_CONSTRUCTED(gmm::sub_interval, SUBJ, 0, 3);
    gmm::copy(gmm::sub_matrix(W, SUBJ, SUBI), A);
    gmm::lu_inverse(A);
    gmm::copy(gmm::transposed(A), gmm::sub_matrix(M, SUBI));

    for (unsigned j = 0; j < 3; ++j) {
      gmm::mult(W, gmm::mat_row(M, j), w);
      gmm::mult(A, gmm::scaled(w, -1.0), coeff);
      gmm::copy(coeff, gmm::sub_vector(gmm::mat_row(M, j), SUBI));
    }
  }

  morley_triangle__::morley_triangle__() {
    cvr = bgeot::simplex_of_reference(2);
    dim_ = cvr->structure()->dim();
    init_cvs_node();
    es_degree = 2;
    is_pol = true;
    is_standard_fem = is_lag = is_equiv = false;
    base_.resize(6);

    std::stringstream s("1 - x - y + 2*x*y;  (x + y + x^2 - 2*x*y - y^2)/2;"
                        "(x + y - x^2 - 2*x*y + y^2)/2;"
                        "((x+y)^2 - x - y)*sqrt(2)/2;  x*(x-1);  y*(y-1);");

    for (unsigned k = 0; k < 6; ++k)
      base_[k] = read_base_poly(2, s);

    add_node(lagrange_dof(2), base_node(0.0, 0.0));
    add_node(lagrange_dof(2), base_node(1.0, 0.0));
    add_node(lagrange_dof(2), base_node(0.0, 1.0));
    add_node(normal_derivative_dof(2), base_node(0.5, 0.5));
    add_node(normal_derivative_dof(2), base_node(0.0, 0.5));
    add_node(normal_derivative_dof(2), base_node(0.5, 0.0));
  }

  static pfem triangle_Morley_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 0, "Bad number of parameters");
    pfem p = std::make_shared<morley_triangle__>();
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*        DISCONTINUOUS PK                                                  */
  /* ******************************************************************** */

  struct PK_discont_ : public PK_fem_ {
  public :

    PK_discont_(dim_type nc, short_type k, scalar_type alpha=scalar_type(0))
      : PK_fem_(nc, k) {
      std::fill(dof_types_.begin(), dof_types_.end(),
                lagrange_nonconforming_dof(nc));

      if (alpha != scalar_type(0)) {
        base_node G =
          gmm::mean_value(cv_node.points().begin(), cv_node.points().end());
        for (size_type i=0; i < cv_node.nb_points(); ++i)
          cv_node.points()[i] = (1-alpha)*cv_node.points()[i] + alpha*G;
        for (size_type d = 0; d < nc; ++d) {
          base_poly S(1,2);
          S[0] = -alpha * G[d] / (1-alpha);
          S[1] = 1. / (1-alpha);
          for (size_type j=0; j < nb_base(0); ++j) {
            base_[j] = bgeot::poly_substitute_var(base_[j],S,d);
          }
        }
      }
    }
  };

  static pfem PK_discontinuous_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 2 || params.size() == 3,
                "Bad number of parameters : "
                << params.size() << " should be 2 or 3.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0 &&
                (params.size() != 3 || params[2].type() == 0),
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    scalar_type alpha = 0.;
    if (params.size() == 3) alpha = params[2].num();
    GMM_ASSERT1(n > 0 && n < 100 && k >= 0 && k <= 150 && alpha >= 0 &&
                alpha < 1 && double(n) == params[0].num()
                && double(k) == params[1].num(), "Bad parameters");
    pfem p = std::make_shared<PK_discont_>(dim_type(n), short_type(k), alpha);
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }


  /* ******************************************************************** */
  /*        PK element with a bubble base fonction                            */
  /* ******************************************************************** */

  struct PK_with_cubic_bubble_ : public PK_fem_ {
    PK_with_cubic_bubble_(dim_type nc, short_type k);
  };

  PK_with_cubic_bubble_::PK_with_cubic_bubble_(dim_type nc, short_type k)
    : PK_fem_(nc, k) {
    unfreeze_cvs_node();
    is_lag = false; es_degree = short_type(nc+1);
    base_node pt(nc);
    size_type j;
    PK_fem_ P1(nc, 1);

    pt.fill(1./(nc+1)); /* barycenter of the convex */

    add_node(bubble1_dof(nc), pt);
    base_.resize(nb_dof(0));

    j = nb_dof(0) - 1;
    base_[j] = base_poly(nc, 0);
    base_[j].one();
    for (size_type i = 0; i < P1.nb_dof(0); i++) base_[j] *= P1.base()[i];
    // cout << "bubble = " << base_[j] << endl;
  }

  static pfem PK_with_cubic_bubble(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies) {
    GMM_ASSERT1(params.size() == 2, "Bad number of parameters : "
                << params.size() << " should be 2.");
    GMM_ASSERT1(params[0].type() == 0 && params[1].type() == 0,
                "Bad type of parameters");
    int n = int(::floor(params[0].num() + 0.01));
    int k = int(::floor(params[1].num() + 0.01));
    GMM_ASSERT1(k < n+1, "dimensions mismatch");
    GMM_ASSERT1(n > 0 && n < 100 && k >= 0 && k <= 150 &&
                double(n) == params[0].num() && double(k) == params[1].num(),
                "Bad parameters");
    pfem p = std::make_shared<PK_with_cubic_bubble_>(dim_type(n),
                                                     short_type(k));
    dependencies.push_back(p->ref_convex(0));
    dependencies.push_back(p->node_tab(0));
    return p;
  }

  /* ******************************************************************** */
  /*        classical fem                                                     */
  /* ******************************************************************** */

  static pfem classical_fem_(bgeot::pgeometric_trans pgt,
                             short_type k, bool complete=false,
                             bool discont=false, scalar_type alpha=0) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(bgeot::pgeometric_trans, pgt_last,0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(short_type, k_last, short_type(-1));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pfem, fem_last, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(char, complete_last, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(char, discont_last, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(scalar_type, alpha_last, 0);

    bool found = false;
    if (pgt_last == pgt && k_last == k && complete_last == complete &&
        discont_last == discont && alpha_last == alpha)
      return fem_last;

    dim_type n = pgt->structure()->dim();
    dim_type nbp = dim_type(pgt->basic_structure()->nb_points());
    std::stringstream name;
    std::string suffix(discont ? "_DISCONTINUOUS" : "");
    GMM_ASSERT2(discont || alpha == scalar_type(0),
                "Cannot use an alpha parameter in continuous elements.");
    std::stringstream arg_;
    if (discont && alpha != scalar_type(0))
      arg_ << "," << alpha;
    std::string arg(arg_.str());

    // Identifying if it is a torus structure
    if (bgeot::is_torus_geom_trans(pgt) && n == 3) n = 2;

    /* Identifying P1-simplexes.                                          */
    if (nbp == n+1)
      if (pgt->basic_structure() == bgeot::simplex_structure(n)) {
        name << "FEM_PK" << suffix << "(" << n << "," << k << arg << ")";
        found = true;
      }

    /* Identifying Q1-parallelepiped.                                     */
    if (!found && nbp == (size_type(1) << n))
      if (pgt->basic_structure() == bgeot::parallelepiped_structure(n)) {
        if (!complete && k == 2 && (n == 2 || n == 3) &&
            pgt->structure() == bgeot::Q2_incomplete_structure(n)) {
          GMM_ASSERT2(alpha == scalar_type(0),
                      "Cannot use an alpha parameter in incomplete Q2"
                      " elements.");
          name << "FEM_Q2_INCOMPLETE" << suffix << "(" << n << ")";
        } else
          name << "FEM_QK" << suffix << "(" << n << "," << k << arg << ")";
        found = true;
      }

    /* Identifying Q1-prisms.                                             */
    if (!found && nbp == 2 * n)
      if (pgt->basic_structure() == bgeot::prism_P1_structure(n)) {
        if (!complete && k == 2 && n == 3 &&
            pgt->structure() == bgeot::prism_incomplete_P2_structure()) {
          GMM_ASSERT2(alpha == scalar_type(0),
                      "Cannot use an alpha parameter in incomplete prism"
                      " elements.");
          name << "FEM_PRISM_INCOMPLETE_P2" << suffix;
        } else
          name << "FEM_PRISM_PK" << suffix << "(" << n << "," << k << arg << ")";
        found = true;
      }

    /* Identifying pyramids.                                              */
    if (!found && nbp == 5)
      if (pgt->basic_structure() == bgeot::pyramid_QK_structure(1)) {
        if (!complete && k == 2 &&
            pgt->structure() == bgeot::pyramid_Q2_incomplete_structure()) {
          GMM_ASSERT2(alpha == scalar_type(0),
                      "Cannot use an alpha parameter in incomplete pyramid"
                      " elements.");
          name << "FEM_PYRAMID_Q2_INCOMPLETE" << suffix;
        } else
          name << "FEM_PYRAMID_QK" << suffix << "(" << k << arg << ")";
        found = true;;
      }

    // To be completed

    GMM_ASSERT1(found, "This element is not taken into account. Contact us");
    fem_last = fem_descriptor(name.str());
    pgt_last = pgt;
    k_last = k;
    complete_last = complete;
    discont_last = discont;
    alpha_last = alpha;
    return fem_last;
  }

  pfem classical_fem(bgeot::pgeometric_trans pgt, short_type k,
                     bool complete) {
    return classical_fem_(pgt, k, complete);
  }

  pfem classical_discontinuous_fem(bgeot::pgeometric_trans pgt, short_type k,
                                   scalar_type alpha, bool complete) {
    return classical_fem_(pgt, k, complete, true, alpha);
  }

  /* ******************************************************************** */
  /*    Naming system                                                     */
  /* ******************************************************************** */

  pfem structured_composite_fem_method(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem PK_composite_hierarch_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem PK_composite_full_hierarch_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem HCT_triangle_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem reduced_HCT_triangle_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem reduced_quadc1p3_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem quadc1p3_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);
  pfem P1bubbletriangle_fem(fem_param_list &params,
        std::vector<dal::pstatic_stored_object> &dependencies);

  struct fem_naming_system : public dal::naming_system<virtual_fem> {
    fem_naming_system() : dal::naming_system<virtual_fem>("FEM") {
      add_suffix("HERMITE", Hermite_fem);
      add_suffix("ARGYRIS", triangle_Argyris_fem);
      add_suffix("MORLEY", triangle_Morley_fem);
      add_suffix("PK", PK_fem);
      add_suffix("QK", QK_fem);
      add_suffix("QK_DISCONTINUOUS", QK_discontinuous_fem);
      add_suffix("PRISM_PK", prism_PK_fem);
      add_suffix("PK_PRISM", prism_PK_fem); // for backwards compatibility
      add_suffix("PK_DISCONTINUOUS", PK_discontinuous_fem);
      add_suffix("PRISM_PK_DISCONTINUOUS", prism_PK_discontinuous_fem);
      add_suffix("PK_PRISM_DISCONTINUOUS", prism_PK_discontinuous_fem); // for backwards compatibility
      add_suffix("PK_WITH_CUBIC_BUBBLE", PK_with_cubic_bubble);
      add_suffix("PRODUCT", product_fem);
      add_suffix("P1_NONCONFORMING", P1_nonconforming_fem);
      add_suffix("P1_BUBBLE_FACE", P1_with_bubble_on_a_face);
      add_suffix("P1_BUBBLE_FACE_LAG", P1_with_bubble_on_a_face_lagrange);
      add_suffix("P1_PIECEWISE_LINEAR_BUBBLE", P1bubbletriangle_fem);
      add_suffix("GEN_HIERARCHICAL", gen_hierarchical_fem);
      add_suffix("PK_HIERARCHICAL", PK_hierarch_fem);
      add_suffix("QK_HIERARCHICAL", QK_hierarch_fem);
      add_suffix("PRISM_PK_HIERARCHICAL", prism_PK_hierarch_fem);
      add_suffix("PK_PRISM_HIERARCHICAL", prism_PK_hierarch_fem); // for backwards compatibility
      add_suffix("STRUCTURED_COMPOSITE", structured_composite_fem_method);
      add_suffix("PK_HIERARCHICAL_COMPOSITE", PK_composite_hierarch_fem);
      add_suffix("PK_FULL_HIERARCHICAL_COMPOSITE",
                 PK_composite_full_hierarch_fem);
      add_suffix("PK_GAUSSLOBATTO1D", PK_GL_fem);
      add_suffix("Q2_INCOMPLETE", Q2_incomplete_fem);
      add_suffix("Q2_INCOMPLETE_DISCONTINUOUS", Q2_incomplete_discontinuous_fem);
      add_suffix("HCT_TRIANGLE", HCT_triangle_fem);
      add_suffix("REDUCED_HCT_TRIANGLE", reduced_HCT_triangle_fem);
      add_suffix("QUADC1_COMPOSITE", quadc1p3_fem);
      add_suffix("REDUCED_QUADC1_COMPOSITE", reduced_quadc1p3_fem);
      add_suffix("RT0", P1_RT0);
      add_suffix("RT0Q", P1_RT0Q);
      add_suffix("NEDELEC", P1_nedelec);
      add_suffix("PYRAMID_QK", pyramid_QK_fem);
      add_suffix("PYRAMID_QK_DISCONTINUOUS", pyramid_QK_disc_fem);
      add_suffix("PYRAMID_LAGRANGE", pyramid_QK_fem); // for backwards compatibility
      add_suffix("PYRAMID_DISCONTINUOUS_LAGRANGE", pyramid_QK_disc_fem); // for backwards compatibility
      add_suffix("PYRAMID_Q2_INCOMPLETE", pyramid_Q2_incomplete_fem);
      add_suffix("PYRAMID_Q2_INCOMPLETE_DISCONTINUOUS",
                 pyramid_Q2_incomplete_disc_fem);
      add_suffix("PRISM_INCOMPLETE_P2", prism_incomplete_P2_fem);
      add_suffix("PRISM_INCOMPLETE_P2_DISCONTINUOUS",
                 prism_incomplete_P2_disc_fem);
    }
  };

  // get a fem descriptor from a string name of a fem.
  pfem fem_descriptor(const std::string &name) {
    size_type i = 0;
    pfem  pf = dal::singleton<fem_naming_system>::instance().method(name, i);
    const_cast<virtual_fem &>(*pf).debug_name()
      = dal::singleton<fem_naming_system>::instance().shorter_name_of_method(pf);
    return pf;
  }

  // get the string name of a fem descriptor.
  std::string name_of_fem(pfem p) {

    auto &instance = dal::singleton<fem_naming_system>::instance();
    auto *p_torus = dynamic_cast<const torus_fem *>(p.get());
    if (p_torus) return instance.shorter_name_of_method(p_torus->get_original_pfem());
    return instance.shorter_name_of_method(p);

    return dal::singleton<fem_naming_system>::instance().
      shorter_name_of_method(p);
  }

  // allows the add of a fem.
  void add_fem_name(std::string name,
                    dal::naming_system<virtual_fem>::pfunction f) {
    dal::singleton<fem_naming_system>::instance().add_suffix(name, f);
  }

  /* ******************************************************************** */
  /*    Aliases functions                                                 */
  /* ******************************************************************** */

  pfem PK_fem(size_type n, short_type k) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pfem, pf, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(size_type, d, size_type(-2));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(short_type, r, short_type(-2));
    if (d != n || r != k) {
      std::stringstream name;
      name << "FEM_PK(" << n << "," << k << ")";
      pf = fem_descriptor(name.str());
      d = n; r = k;
    }
    return pf;
  }

  pfem QK_fem(size_type n, short_type k) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pfem, pf, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(size_type, d, size_type(-2));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(short_type, r, short_type(-2));
    if (d != n || r != k) {
      std::stringstream name;
      name << "FEM_QK(" << n << "," << k << ")";
      pf = fem_descriptor(name.str());
      d = n; r = k;
    }
    return pf;
  }

  pfem prism_PK_fem(size_type n, short_type k) {
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(pfem, pf, 0);
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(size_type, d, size_type(-2));
    DEFINE_STATIC_THREAD_LOCAL_INITIALIZED(short_type, r, short_type(-2));
    if (d != n || r != k) {
      std::stringstream name;
      name << "FEM_PRISM_PK(" << n << "," << k << ")";
      pf = fem_descriptor(name.str());
      d = n; r = k;
    }
    return pf;
  }


  /* ********************************************************************* */
  /*       Precomputation on fem.                                          */
  /* ********************************************************************* */

  DAL_DOUBLE_KEY(pre_fem_key_, pfem, bgeot::pstored_point_tab);

  fem_precomp_::fem_precomp_(const pfem pff, const bgeot::pstored_point_tab ps) :
    pf(pff), pspt(ps) {
    DAL_STORED_OBJECT_DEBUG_CREATED(this, "Fem_precomp");
    for (const auto &p : *pspt)
      GMM_ASSERT1(p.size() == pf->dim(), "dimensions mismatch");
  }

  void fem_precomp_::init_val() const {
    c.resize(pspt->size());
    for (size_type i = 0; i < pspt->size(); ++i)
      pf->base_value((*pspt)[i], c[i]);
  }

  void fem_precomp_::init_grad() const {
    pc.resize(pspt->size());
    for (size_type i = 0; i < pspt->size(); ++i)
      pf->grad_base_value((*pspt)[i], pc[i]);
  }

  void fem_precomp_::init_hess() const {
    hpc.resize(pspt->size());
    for (size_type i = 0; i < pspt->size(); ++i)
      pf->hess_base_value((*pspt)[i], hpc[i]);
  }

  pfem_precomp fem_precomp(pfem pf, bgeot::pstored_point_tab pspt,
                           dal::pstatic_stored_object dep) {
    dal::pstatic_stored_object_key pk = std::make_shared<pre_fem_key_>(pf,pspt);
    dal::pstatic_stored_object o = dal::search_stored_object(pk);
    if (o) return std::dynamic_pointer_cast<const fem_precomp_>(o);
    pfem_precomp p = std::make_shared<fem_precomp_>(pf, pspt);
    dal::add_stored_object(pk, p, pspt, dal::AUTODELETE_STATIC_OBJECT);
    if (dal::exists_stored_object(pf)) dal::add_dependency(p, pf);
    if (dep) dal::add_dependency(p, dep);
    return p;
  }

  void fem_precomp_pool::clear() {
    for (const pfem_precomp &p : precomps)
      del_stored_object(p, true);
    precomps.clear();
  }


}  /* end of namespace getfem.                                            */