nonlinearform.cpp

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// Copyright (c) 2010-2024, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
 
#include "fem.hpp"
#include "../general/forall.hpp"
 
namespace mfem
{
 
void NonlinearForm::SetAssemblyLevel(AssemblyLevel assembly_level)
{
   if (ext)
   {
      MFEM_ABORT("the assembly level has already been set!");
   }
   assembly = assembly_level;
   switch (assembly)
   {
      case AssemblyLevel::NONE:
         ext = new MFNonlinearFormExtension(this);
         break;
      case AssemblyLevel::PARTIAL:
         ext = new PANonlinearFormExtension(this);
         break;
      case AssemblyLevel::LEGACY:
         // This is the default
         break;
      default:
         mfem_error("Unknown assembly level for this form.");
   }
}
 
void NonlinearForm::SetEssentialBC(const Array<int> &bdr_attr_is_ess,
                                   Vector *rhs)
{
   // virtual call, works in parallel too
   fes->GetEssentialTrueDofs(bdr_attr_is_ess, ess_tdof_list);
 
   if (rhs)
   {
      for (int i = 0; i < ess_tdof_list.Size(); i++)
      {
         (*rhs)(ess_tdof_list[i]) = 0.0;
      }
   }
}
 
void NonlinearForm::SetEssentialVDofs(const Array<int> &ess_vdofs_list)
{
   if (!P)
   {
      ess_vdofs_list.Copy(ess_tdof_list); // ess_vdofs_list --> ess_tdof_list
   }
   else
   {
      Array<int> ess_vdof_marker, ess_tdof_marker;
      FiniteElementSpace::ListToMarker(ess_vdofs_list, fes->GetVSize(),
                                       ess_vdof_marker);
      if (Serial())
      {
         fes->ConvertToConformingVDofs(ess_vdof_marker, ess_tdof_marker);
      }
      else
      {
#ifdef MFEM_USE_MPI
         ParFiniteElementSpace *pf = dynamic_cast<ParFiniteElementSpace*>(fes);
         ess_tdof_marker.SetSize(pf->GetTrueVSize());
         pf->Dof_TrueDof_Matrix()->BooleanMultTranspose(1, ess_vdof_marker,
                                                        0, ess_tdof_marker);
#else
         MFEM_ABORT("internal MFEM error");
#endif
      }
      FiniteElementSpace::MarkerToList(ess_tdof_marker, ess_tdof_list);
   }
}
 
real_t NonlinearForm::GetGridFunctionEnergy(const Vector &x) const
{
   if (ext)
   {
      MFEM_VERIFY(!fnfi.Size(), "Interior faces terms not yet implemented!");
      MFEM_VERIFY(!bfnfi.Size(), "Boundary face terms not yet implemented!");
      return ext->GetGridFunctionEnergy(x);
   }
 
   Array<int> vdofs;
   Vector el_x;
   const FiniteElement *fe;
   ElementTransformation *T;
   DofTransformation *doftrans;
   Mesh *mesh = fes->GetMesh();
   real_t energy = 0.0;
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNE(); i++)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         fe = fes->GetFE(i);
         doftrans = fes->GetElementVDofs(i, vdofs);
         T = fes->GetElementTransformation(i);
         x.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < dnfi.Size(); k++)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            energy += dnfi[k]->GetElementEnergy(*fe, *T, el_x);
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         fe = fes->GetBE(i);
         doftrans = fes->GetBdrElementVDofs(i, vdofs);
         T = fes->GetBdrElementTransformation(i);
         x.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            energy += bnfi[k]->GetElementEnergy(*fe, *T, el_x);
         }
      }
 
   }
 
   if (fnfi.Size())
   {
      MFEM_ABORT("TODO: add energy contribution from interior face terms");
   }
 
   if (bfnfi.Size())
   {
      MFEM_ABORT("TODO: add energy contribution from boundary face terms");
   }
 
   return energy;
}
 
const Vector &NonlinearForm::Prolongate(const Vector &x) const
{
   MFEM_VERIFY(x.Size() == Width(), "invalid input Vector size");
   if (P)
   {
      aux1.SetSize(P->Height());
      P->Mult(x, aux1);
      return aux1;
   }
   return x;
}
 
void NonlinearForm::Mult(const Vector &x, Vector &y) const
{
   const Vector &px = Prolongate(x);
   if (P) { aux2.SetSize(P->Height()); }
 
   // If we are in parallel, ParNonLinearForm::Mult uses the aux2 vector. In
   // serial, place the result directly in y (when there is no P).
   Vector &py = P ? aux2 : y;
 
   if (ext)
   {
      ext->Mult(px, py);
      if (Serial())
      {
         if (cP) { cP->MultTranspose(py, y); }
         const int N = ess_tdof_list.Size();
         const auto tdof = ess_tdof_list.Read();
         auto Y = y.ReadWrite();
         mfem::forall(N, [=] MFEM_HOST_DEVICE (int i) { Y[tdof[i]] = 0.0; });
      }
      // In parallel, the result is in 'py' which is an alias for 'aux2'.
      return;
   }
 
   Array<int> vdofs;
   Vector el_x, el_y;
   const FiniteElement *fe;
   ElementTransformation *T;
   DofTransformation *doftrans;
   Mesh *mesh = fes->GetMesh();
 
   py = 0.0;
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNE(); i++)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         fe = fes->GetFE(i);
         doftrans = fes->GetElementVDofs(i, vdofs);
         T = fes->GetElementTransformation(i);
         px.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < dnfi.Size(); k++)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            dnfi[k]->AssembleElementVector(*fe, *T, el_x, el_y);
            if (doftrans) {doftrans->TransformDual(el_y); }
            py.AddElementVector(vdofs, el_y);
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         fe = fes->GetBE(i);
         doftrans = fes->GetBdrElementVDofs(i, vdofs);
         T = fes->GetBdrElementTransformation(i);
         px.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            bnfi[k]->AssembleElementVector(*fe, *T, el_x, el_y);
            if (doftrans) {doftrans->TransformDual(el_y); }
            py.AddElementVector(vdofs, el_y);
         }
      }
   }
 
   if (fnfi.Size())
   {
      FaceElementTransformations *tr;
      const FiniteElement *fe1, *fe2;
      Array<int> vdofs2;
 
      for (int i = 0; i < mesh->GetNumFaces(); i++)
      {
         tr = mesh->GetInteriorFaceTransformations(i);
         if (tr != NULL)
         {
            fes->GetElementVDofs(tr->Elem1No, vdofs);
            fes->GetElementVDofs(tr->Elem2No, vdofs2);
            vdofs.Append (vdofs2);
 
            px.GetSubVector(vdofs, el_x);
 
            fe1 = fes->GetFE(tr->Elem1No);
            fe2 = fes->GetFE(tr->Elem2No);
 
            for (int k = 0; k < fnfi.Size(); k++)
            {
               fnfi[k]->AssembleFaceVector(*fe1, *fe2, *tr, el_x, el_y);
               py.AddElementVector(vdofs, el_y);
            }
         }
      }
   }
 
   if (bfnfi.Size())
   {
      FaceElementTransformations *tr;
      const FiniteElement *fe1, *fe2;
 
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bfnfi.Size(); k++)
      {
         if (bfnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bfnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary face integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < fes -> GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         tr = mesh->GetBdrFaceTransformations (i);
         if (tr != NULL)
         {
            fes->GetElementVDofs(tr->Elem1No, vdofs);
            px.GetSubVector(vdofs, el_x);
 
            fe1 = fes->GetFE(tr->Elem1No);
            // The fe2 object is really a dummy and not used on the boundaries,
            // but we can't dereference a NULL pointer, and we don't want to
            // actually make a fake element.
            fe2 = fe1;
            for (int k = 0; k < bfnfi.Size(); k++)
            {
               if (bfnfi_marker[k] &&
                   (*bfnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
               bfnfi[k]->AssembleFaceVector(*fe1, *fe2, *tr, el_x, el_y);
               py.AddElementVector(vdofs, el_y);
            }
         }
      }
   }
 
   if (Serial())
   {
      if (cP) { cP->MultTranspose(py, y); }
 
      for (int i = 0; i < ess_tdof_list.Size(); i++)
      {
         y(ess_tdof_list[i]) = 0.0;
      }
      // y(ess_tdof_list[i]) = x(ess_tdof_list[i]);
   }
   // In parallel, the result is in 'py' which is an alias for 'aux2'.
}
 
Operator &NonlinearForm::GetGradient(const Vector &x) const
{
   if (ext)
   {
      hGrad.Clear();
      Operator &grad = ext->GetGradient(Prolongate(x));
      Operator *Gop;
      grad.FormSystemOperator(ess_tdof_list, Gop);
      hGrad.Reset(Gop);
      // In both serial and parallel, when using extension, we return the final
      // global true-dof gradient with imposed b.c.
      return *hGrad;
   }
 
   const int skip_zeros = 0;
   Array<int> vdofs;
   Vector el_x;
   DenseMatrix elmat;
   const FiniteElement *fe;
   ElementTransformation *T;
   DofTransformation *doftrans;
   Mesh *mesh = fes->GetMesh();
   const Vector &px = Prolongate(x);
 
   if (Grad == NULL)
   {
      Grad = new SparseMatrix(fes->GetVSize());
   }
   else
   {
      *Grad = 0.0;
   }
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNE(); i++)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         fe = fes->GetFE(i);
         doftrans = fes->GetElementVDofs(i, vdofs);
         T = fes->GetElementTransformation(i);
         px.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < dnfi.Size(); k++)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            dnfi[k]->AssembleElementGrad(*fe, *T, el_x, elmat);
            if (doftrans) { doftrans->TransformDual(elmat); }
            Grad->AddSubMatrix(vdofs, vdofs, elmat, skip_zeros);
            // Grad->AddSubMatrix(vdofs, vdofs, elmat, 1);
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < fes->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         fe = fes->GetBE(i);
         doftrans = fes->GetBdrElementVDofs(i, vdofs);
         T = fes->GetBdrElementTransformation(i);
         px.GetSubVector(vdofs, el_x);
         if (doftrans) {doftrans->InvTransformPrimal(el_x); }
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            bnfi[k]->AssembleElementGrad(*fe, *T, el_x, elmat);
            if (doftrans) { doftrans->TransformDual(elmat); }
            Grad->AddSubMatrix(vdofs, vdofs, elmat, skip_zeros);
         }
      }
   }
 
   if (fnfi.Size())
   {
      FaceElementTransformations *tr;
      const FiniteElement *fe1, *fe2;
      Array<int> vdofs2;
 
      for (int i = 0; i < mesh->GetNumFaces(); i++)
      {
         tr = mesh->GetInteriorFaceTransformations(i);
         if (tr != NULL)
         {
            fes->GetElementVDofs(tr->Elem1No, vdofs);
            fes->GetElementVDofs(tr->Elem2No, vdofs2);
            vdofs.Append (vdofs2);
 
            px.GetSubVector(vdofs, el_x);
 
            fe1 = fes->GetFE(tr->Elem1No);
            fe2 = fes->GetFE(tr->Elem2No);
 
            for (int k = 0; k < fnfi.Size(); k++)
            {
               fnfi[k]->AssembleFaceGrad(*fe1, *fe2, *tr, el_x, elmat);
               Grad->AddSubMatrix(vdofs, vdofs, elmat, skip_zeros);
            }
         }
      }
   }
 
   if (bfnfi.Size())
   {
      FaceElementTransformations *tr;
      const FiniteElement *fe1, *fe2;
 
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bfnfi.Size(); k++)
      {
         if (bfnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bfnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary face integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < fes -> GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         tr = mesh->GetBdrFaceTransformations (i);
         if (tr != NULL)
         {
            fes->GetElementVDofs(tr->Elem1No, vdofs);
            px.GetSubVector(vdofs, el_x);
 
            fe1 = fes->GetFE(tr->Elem1No);
            // The fe2 object is really a dummy and not used on the boundaries,
            // but we can't dereference a NULL pointer, and we don't want to
            // actually make a fake element.
            fe2 = fe1;
            for (int k = 0; k < bfnfi.Size(); k++)
            {
               if (bfnfi_marker[k] &&
                   (*bfnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
               bfnfi[k]->AssembleFaceGrad(*fe1, *fe2, *tr, el_x, elmat);
               Grad->AddSubMatrix(vdofs, vdofs, elmat, skip_zeros);
            }
         }
      }
   }
 
   if (!Grad->Finalized())
   {
      Grad->Finalize(skip_zeros);
   }
 
   SparseMatrix *mGrad = Grad;
   if (Serial())
   {
      if (cP)
      {
         delete cGrad;
         cGrad = RAP(*cP, *Grad, *cP);
         mGrad = cGrad;
      }
      for (int i = 0; i < ess_tdof_list.Size(); i++)
      {
         mGrad->EliminateRowCol(ess_tdof_list[i]);
      }
   }
 
   return *mGrad;
}
 
void NonlinearForm::Update()
{
   if (sequence == fes->GetSequence()) { return; }
 
   height = width = fes->GetTrueVSize();
   delete cGrad; cGrad = NULL;
   delete Grad; Grad = NULL;
   hGrad.Clear();
   ess_tdof_list.SetSize(0); // essential b.c. will need to be set again
   sequence = fes->GetSequence();
   // Do not modify aux1 and aux2, their size will be set before use.
   P = fes->GetProlongationMatrix();
   cP = dynamic_cast<const SparseMatrix*>(P);
 
   if (ext) { ext->Update(); }
}
 
void NonlinearForm::Setup()
{
   if (ext) { ext->Assemble(); }
}
 
NonlinearForm::~NonlinearForm()
{
   delete cGrad;
   delete Grad;
   if (!extern_bfs)
   {
      for (int i = 0; i <  dnfi.Size(); i++) { delete  dnfi[i]; }
      for (int i = 0; i <  bnfi.Size(); i++) { delete  bnfi[i]; }
      for (int i = 0; i <  fnfi.Size(); i++) { delete  fnfi[i]; }
      for (int i = 0; i < bfnfi.Size(); i++) { delete bfnfi[i]; }
   }
   delete ext;
}
 
 
BlockNonlinearForm::BlockNonlinearForm() :
   fes(0), BlockGrad(NULL)
{
   height = 0;
   width = 0;
}
 
void BlockNonlinearForm::SetSpaces(Array<FiniteElementSpace *> &f)
{
   delete BlockGrad;
   BlockGrad = NULL;
   for (int i=0; i<Grads.NumRows(); ++i)
   {
      for (int j=0; j<Grads.NumCols(); ++j)
      {
         delete Grads(i,j);
         delete cGrads(i,j);
      }
   }
   for (int i = 0; i < ess_tdofs.Size(); ++i)
   {
      delete ess_tdofs[i];
   }
 
   height = 0;
   width = 0;
   f.Copy(fes);
   block_offsets.SetSize(f.Size() + 1);
   block_trueOffsets.SetSize(f.Size() + 1);
   block_offsets[0] = 0;
   block_trueOffsets[0] = 0;
 
   for (int i=0; i<fes.Size(); ++i)
   {
      block_offsets[i+1] = fes[i]->GetVSize();
      block_trueOffsets[i+1] = fes[i]->GetTrueVSize();
   }
 
   block_offsets.PartialSum();
   block_trueOffsets.PartialSum();
 
   height = block_trueOffsets[fes.Size()];
   width = block_trueOffsets[fes.Size()];
 
   Grads.SetSize(fes.Size(), fes.Size());
   Grads = NULL;
 
   cGrads.SetSize(fes.Size(), fes.Size());
   cGrads = NULL;
 
   P.SetSize(fes.Size());
   cP.SetSize(fes.Size());
   ess_tdofs.SetSize(fes.Size());
   for (int s = 0; s < fes.Size(); ++s)
   {
      // Retrieve prolongation matrix for each FE space
      P[s] = fes[s]->GetProlongationMatrix();
      cP[s] = dynamic_cast<const SparseMatrix *>(P[s]);
 
      // If the P Operator exists and its type is not SparseMatrix, this
      // indicates the Operator is part of parallel run.
      if (P[s] && !cP[s])
      {
         is_serial = false;
      }
 
      // If the P Operator exists and its type is SparseMatrix, this indicates
      // the Operator is serial but needs prolongation on assembly.
      if (cP[s])
      {
         needs_prolongation = true;
      }
 
      ess_tdofs[s] = new Array<int>;
   }
}
 
BlockNonlinearForm::BlockNonlinearForm(Array<FiniteElementSpace *> &f) :
   fes(0), BlockGrad(NULL)
{
   SetSpaces(f);
}
 
void BlockNonlinearForm::SetEssentialBC(
   const Array<Array<int> *> &bdr_attr_is_ess, Array<Vector *> &rhs)
{
   for (int s = 0; s < fes.Size(); ++s)
   {
      ess_tdofs[s]->SetSize(ess_tdofs.Size());
 
      fes[s]->GetEssentialTrueDofs(*bdr_attr_is_ess[s], *ess_tdofs[s]);
 
      if (rhs[s])
      {
         rhs[s]->SetSubVector(*ess_tdofs[s], 0.0);
      }
   }
}
 
real_t BlockNonlinearForm::GetEnergyBlocked(const BlockVector &bx) const
{
   Array<Array<int> *> vdofs(fes.Size());
   Array<Vector *> el_x(fes.Size());
   Array<const Vector *> el_x_const(fes.Size());
   Array<const FiniteElement *> fe(fes.Size());
   ElementTransformation *T;
   DofTransformation *doftrans;
   Mesh *mesh = fes[0]->GetMesh();
   real_t energy = 0.0;
 
   for (int i=0; i<fes.Size(); ++i)
   {
      el_x_const[i] = el_x[i] = new Vector();
      vdofs[i] = new Array<int>;
   }
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes[0]->GetNE(); ++i)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         T = fes[0]->GetElementTransformation(i);
         for (int s=0; s<fes.Size(); ++s)
         {
            fe[s] = fes[s]->GetFE(i);
            doftrans = fes[s]->GetElementVDofs(i, *vdofs[s]);
            bx.GetBlock(s).GetSubVector(*vdofs[s], *el_x[s]);
            if (doftrans) {doftrans->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < dnfi.Size(); ++k)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            energy += dnfi[k]->GetElementEnergy(fe, *T, el_x_const);
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < mesh->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         T = fes[0]->GetBdrElementTransformation(i);
         for (int s = 0; s < fes.Size(); ++s)
         {
            fe[s] = fes[s]->GetBE(i);
            doftrans = fes[s]->GetBdrElementVDofs(i, *(vdofs[s]));
            bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            if (doftrans) {doftrans->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            energy += bnfi[k]->GetElementEnergy(fe, *T, el_x_const);
         }
      }
   }
 
   // free the allocated memory
   for (int i = 0; i < fes.Size(); ++i)
   {
      delete el_x[i];
      delete vdofs[i];
   }
 
   if (fnfi.Size())
   {
      MFEM_ABORT("TODO: add energy contribution from interior face terms");
   }
 
   if (bfnfi.Size())
   {
      MFEM_ABORT("TODO: add energy contribution from boundary face terms");
   }
 
   return energy;
}
 
real_t BlockNonlinearForm::GetEnergy(const Vector &x) const
{
   xs.Update(const_cast<Vector&>(x), block_offsets);
   return GetEnergyBlocked(xs);
}
 
void BlockNonlinearForm::MultBlocked(const BlockVector &bx,
                                     BlockVector &by) const
{
   Array<Array<int> *>vdofs(fes.Size());
   Array<Array<int> *>vdofs2(fes.Size());
   Array<Vector *> el_x(fes.Size());
   Array<const Vector *> el_x_const(fes.Size());
   Array<Vector *> el_y(fes.Size());
   Array<const FiniteElement *> fe(fes.Size());
   Array<const FiniteElement *> fe2(fes.Size());
   ElementTransformation *T;
   Array<DofTransformation *> doftrans(fes.Size()); doftrans = nullptr;
   Mesh *mesh = fes[0]->GetMesh();
 
   by.UseDevice(true);
   by = 0.0;
   by.SyncToBlocks();
   for (int s=0; s<fes.Size(); ++s)
   {
      el_x_const[s] = el_x[s] = new Vector();
      el_y[s] = new Vector();
      vdofs[s] = new Array<int>;
      vdofs2[s] = new Array<int>;
   }
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes[0]->GetNE(); ++i)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         T = fes[0]->GetElementTransformation(i);
         for (int s = 0; s < fes.Size(); ++s)
         {
            doftrans[s] = fes[s]->GetElementVDofs(i, *(vdofs[s]));
            fe[s] = fes[s]->GetFE(i);
            bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            if (doftrans[s]) {doftrans[s]->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < dnfi.Size(); ++k)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            dnfi[k]->AssembleElementVector(fe, *T,
                                           el_x_const, el_y);
 
            for (int s=0; s<fes.Size(); ++s)
            {
               if (el_y[s]->Size() == 0) { continue; }
               if (doftrans[s]) {doftrans[s]->TransformDual(*el_y[s]); }
               by.GetBlock(s).AddElementVector(*(vdofs[s]), *el_y[s]);
            }
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < mesh->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         T = fes[0]->GetBdrElementTransformation(i);
         for (int s = 0; s < fes.Size(); ++s)
         {
            doftrans[s] = fes[s]->GetBdrElementVDofs(i, *(vdofs[s]));
            fe[s] = fes[s]->GetBE(i);
            bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            if (doftrans[s]) {doftrans[s]->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            bnfi[k]->AssembleElementVector(fe, *T, el_x_const, el_y);
 
            for (int s=0; s<fes.Size(); ++s)
            {
               if (el_y[s]->Size() == 0) { continue; }
               if (doftrans[s]) {doftrans[s]->TransformDual(*el_y[s]); }
               by.GetBlock(s).AddElementVector(*(vdofs[s]), *el_y[s]);
            }
         }
      }
   }
 
 
   if (fnfi.Size())
   {
      FaceElementTransformations *tr;
 
      for (int i = 0; i < mesh->GetNumFaces(); ++i)
      {
         tr = mesh->GetInteriorFaceTransformations(i);
         if (tr != NULL)
         {
            for (int s=0; s<fes.Size(); ++s)
            {
               fe[s] = fes[s]->GetFE(tr->Elem1No);
               fe2[s] = fes[s]->GetFE(tr->Elem2No);
 
               fes[s]->GetElementVDofs(tr->Elem1No, *(vdofs[s]));
               fes[s]->GetElementVDofs(tr->Elem2No, *(vdofs2[s]));
 
               vdofs[s]->Append(*(vdofs2[s]));
 
               bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            }
 
            for (int k = 0; k < fnfi.Size(); ++k)
            {
 
               fnfi[k]->AssembleFaceVector(fe, fe2, *tr, el_x_const, el_y);
 
               for (int s=0; s<fes.Size(); ++s)
               {
                  if (el_y[s]->Size() == 0) { continue; }
                  by.GetBlock(s).AddElementVector(*(vdofs[s]), *el_y[s]);
               }
            }
         }
      }
   }
 
   if (bfnfi.Size())
   {
      FaceElementTransformations *tr;
 
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bfnfi.Size(); ++k)
      {
         if (bfnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bfnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary face integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); ++i)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < mesh->GetNBE(); ++i)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         tr = mesh->GetBdrFaceTransformations(i);
         if (tr != NULL)
         {
            for (int s=0; s<fes.Size(); ++s)
            {
               fe[s] = fes[s]->GetFE(tr->Elem1No);
               fe2[s] = fes[s]->GetFE(tr->Elem1No);
 
               fes[s]->GetElementVDofs(tr->Elem1No, *(vdofs[s]));
               bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            }
 
            for (int k = 0; k < bfnfi.Size(); ++k)
            {
               if (bfnfi_marker[k] &&
                   (*bfnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
               bfnfi[k]->AssembleFaceVector(fe, fe2, *tr, el_x_const, el_y);
 
               for (int s=0; s<fes.Size(); ++s)
               {
                  if (el_y[s]->Size() == 0) { continue; }
                  by.GetBlock(s).AddElementVector(*(vdofs[s]), *el_y[s]);
               }
            }
         }
      }
   }
 
   for (int s=0; s<fes.Size(); ++s)
   {
      delete vdofs2[s];
      delete vdofs[s];
      delete el_y[s];
      delete el_x[s];
   }
 
   by.SyncFromBlocks();
}
 
const BlockVector &BlockNonlinearForm::Prolongate(const BlockVector &bx) const
{
   MFEM_VERIFY(bx.Size() == Width(), "invalid input BlockVector size");
 
   if (needs_prolongation)
   {
      aux1.Update(block_offsets);
      for (int s = 0; s < fes.Size(); s++)
      {
         P[s]->Mult(bx.GetBlock(s), aux1.GetBlock(s));
      }
      return aux1;
   }
   return bx;
}
 
void BlockNonlinearForm::Mult(const Vector &x, Vector &y) const
{
   BlockVector bx(const_cast<Vector&>(x), block_trueOffsets);
   BlockVector by(y, block_trueOffsets);
 
   const BlockVector &pbx = Prolongate(bx);
   if (needs_prolongation)
   {
      aux2.Update(block_offsets);
   }
   BlockVector &pby = needs_prolongation ? aux2 : by;
 
   xs.Update(const_cast<BlockVector&>(pbx), block_offsets);
   ys.Update(pby, block_offsets);
   MultBlocked(xs, ys);
 
   for (int s = 0; s < fes.Size(); s++)
   {
      if (cP[s])
      {
         cP[s]->MultTranspose(pby.GetBlock(s), by.GetBlock(s));
      }
      by.GetBlock(s).SetSubVector(*ess_tdofs[s], 0.0);
   }
}
 
void BlockNonlinearForm::ComputeGradientBlocked(const BlockVector &bx) const
{
   const int skip_zeros = 0;
   Array<Array<int> *> vdofs(fes.Size());
   Array<Array<int> *> vdofs2(fes.Size());
   Array<Vector *> el_x(fes.Size());
   Array<const Vector *> el_x_const(fes.Size());
   Array2D<DenseMatrix *> elmats(fes.Size(), fes.Size());
   Array<const FiniteElement *>fe(fes.Size());
   Array<const FiniteElement *>fe2(fes.Size());
   ElementTransformation * T;
   Array<DofTransformation *> doftrans(fes.Size()); doftrans = nullptr;
   Mesh *mesh = fes[0]->GetMesh();
 
   for (int i=0; i<fes.Size(); ++i)
   {
      el_x_const[i] = el_x[i] = new Vector();
      vdofs[i] = new Array<int>;
      vdofs2[i] = new Array<int>;
      for (int j=0; j<fes.Size(); ++j)
      {
         elmats(i,j) = new DenseMatrix();
      }
   }
 
   for (int i=0; i<fes.Size(); ++i)
   {
      for (int j=0; j<fes.Size(); ++j)
      {
         if (Grads(i,j) != NULL)
         {
            *Grads(i,j) = 0.0;
         }
         else
         {
            Grads(i,j) = new SparseMatrix(fes[i]->GetVSize(),
                                          fes[j]->GetVSize());
         }
      }
   }
 
   if (dnfi.Size())
   {
      // Which attributes need to be processed?
      Array<int> attr_marker(mesh->attributes.Size() ?
                             mesh->attributes.Max() : 0);
      attr_marker = 0;
      for (int k = 0; k < dnfi.Size(); k++)
      {
         if (dnfi_marker[k] == NULL)
         {
            attr_marker = 1;
            break;
         }
         Array<int> &marker = *dnfi_marker[k];
         MFEM_ASSERT(marker.Size() == attr_marker.Size(),
                     "invalid marker for domain integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < attr_marker.Size(); i++)
         {
            attr_marker[i] |= marker[i];
         }
      }
 
      for (int i = 0; i < fes[0]->GetNE(); ++i)
      {
         const int attr = mesh->GetAttribute(i);
         if (attr_marker[attr-1] == 0) { continue; }
 
         T = fes[0]->GetElementTransformation(i);
         for (int s = 0; s < fes.Size(); ++s)
         {
            fe[s] = fes[s]->GetFE(i);
            doftrans[s] = fes[s]->GetElementVDofs(i, *vdofs[s]);
            bx.GetBlock(s).GetSubVector(*vdofs[s], *el_x[s]);
            if (doftrans[s]) {doftrans[s]->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < dnfi.Size(); ++k)
         {
            if (dnfi_marker[k] &&
                (*dnfi_marker[k])[attr-1] == 0) { continue; }
 
            dnfi[k]->AssembleElementGrad(fe, *T, el_x_const, elmats);
 
            for (int j=0; j<fes.Size(); ++j)
            {
               for (int l=0; l<fes.Size(); ++l)
               {
                  if (elmats(j,l)->Height() == 0) { continue; }
                  if (doftrans[j] || doftrans[l])
                  {
                     TransformDual(doftrans[j], doftrans[l], *elmats(j,l));
                  }
                  Grads(j,l)->AddSubMatrix(*vdofs[j], *vdofs[l],
                                           *elmats(j,l), skip_zeros);
               }
            }
         }
      }
   }
 
   if (bnfi.Size())
   {
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bnfi.Size(); k++)
      {
         if (bnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); i++)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < mesh->GetNBE(); i++)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         T = fes[0]->GetBdrElementTransformation(i);
         for (int s = 0; s < fes.Size(); ++s)
         {
            fe[s] = fes[s]->GetBE(i);
            doftrans[s] = fes[s]->GetBdrElementVDofs(i, *(vdofs[s]));
            bx.GetBlock(s).GetSubVector(*(vdofs[s]), *el_x[s]);
            if (doftrans[s]) {doftrans[s]->InvTransformPrimal(*el_x[s]); }
         }
 
         for (int k = 0; k < bnfi.Size(); k++)
         {
            if (bnfi_marker[k] &&
                (*bnfi_marker[k])[bdr_attr-1] == 0) { continue; }
 
            bnfi[k]->AssembleElementGrad(fe, *T, el_x_const, elmats);
 
            for (int j=0; j<fes.Size(); ++j)
            {
               for (int l=0; l<fes.Size(); ++l)
               {
                  if (elmats(j,l)->Height() == 0) { continue; }
                  if (doftrans[j] || doftrans[l])
                  {
                     TransformDual(doftrans[j], doftrans[l], *elmats(j,l));
                  }
                  Grads(j,l)->AddSubMatrix(*vdofs[j], *vdofs[l],
                                           *elmats(j,l), skip_zeros);
               }
            }
         }
      }
   }
 
   if (fnfi.Size())
   {
      FaceElementTransformations *tr;
 
      for (int i = 0; i < mesh->GetNumFaces(); ++i)
      {
         tr = mesh->GetInteriorFaceTransformations(i);
 
         for (int s=0; s < fes.Size(); ++s)
         {
            fe[s] = fes[s]->GetFE(tr->Elem1No);
            fe2[s] = fes[s]->GetFE(tr->Elem2No);
 
            fes[s]->GetElementVDofs(tr->Elem1No, *vdofs[s]);
            fes[s]->GetElementVDofs(tr->Elem2No, *vdofs2[s]);
            vdofs[s]->Append(*(vdofs2[s]));
 
            bx.GetBlock(s).GetSubVector(*vdofs[s], *el_x[s]);
         }
 
         for (int k = 0; k < fnfi.Size(); ++k)
         {
            fnfi[k]->AssembleFaceGrad(fe, fe2, *tr, el_x_const, elmats);
            for (int j=0; j<fes.Size(); ++j)
            {
               for (int l=0; l<fes.Size(); ++l)
               {
                  if (elmats(j,l)->Height() == 0) { continue; }
                  Grads(j,l)->AddSubMatrix(*vdofs[j], *vdofs[l],
                                           *elmats(j,l), skip_zeros);
               }
            }
         }
      }
   }
 
   if (bfnfi.Size())
   {
      FaceElementTransformations *tr;
 
      // Which boundary attributes need to be processed?
      Array<int> bdr_attr_marker(mesh->bdr_attributes.Size() ?
                                 mesh->bdr_attributes.Max() : 0);
      bdr_attr_marker = 0;
      for (int k = 0; k < bfnfi.Size(); ++k)
      {
         if (bfnfi_marker[k] == NULL)
         {
            bdr_attr_marker = 1;
            break;
         }
         Array<int> &bdr_marker = *bfnfi_marker[k];
         MFEM_ASSERT(bdr_marker.Size() == bdr_attr_marker.Size(),
                     "invalid boundary marker for boundary face integrator #"
                     << k << ", counting from zero");
         for (int i = 0; i < bdr_attr_marker.Size(); ++i)
         {
            bdr_attr_marker[i] |= bdr_marker[i];
         }
      }
 
      for (int i = 0; i < mesh->GetNBE(); ++i)
      {
         const int bdr_attr = mesh->GetBdrAttribute(i);
         if (bdr_attr_marker[bdr_attr-1] == 0) { continue; }
 
         tr = mesh->GetBdrFaceTransformations(i);
         if (tr != NULL)
         {
            for (int s = 0; s < fes.Size(); ++s)
            {
               fe[s] = fes[s]->GetFE(tr->Elem1No);
               fe2[s] = fe[s];
 
               fes[s]->GetElementVDofs(tr->Elem1No, *vdofs[s]);
               bx.GetBlock(s).GetSubVector(*vdofs[s], *el_x[s]);
            }
 
            for (int k = 0; k < bfnfi.Size(); ++k)
            {
               if (bfnfi_marker[k] &&
                   (*bfnfi_marker[k])[bdr_attr-1] == 0) { continue; }
               bfnfi[k]->AssembleFaceGrad(fe, fe2, *tr, el_x_const, elmats);
               for (int l=0; l<fes.Size(); ++l)
               {
                  for (int j=0; j<fes.Size(); ++j)
                  {
                     if (elmats(j,l)->Height() == 0) { continue; }
                     Grads(j,l)->AddSubMatrix(*vdofs[j], *vdofs[l],
                                              *elmats(j,l), skip_zeros);
                  }
               }
            }
         }
      }
   }
 
   if (!Grads(0,0)->Finalized())
   {
      for (int i=0; i<fes.Size(); ++i)
      {
         for (int j=0; j<fes.Size(); ++j)
         {
            Grads(i,j)->Finalize(skip_zeros);
         }
      }
   }
 
   for (int i=0; i<fes.Size(); ++i)
   {
      for (int j=0; j<fes.Size(); ++j)
      {
         delete elmats(i,j);
      }
      delete vdofs2[i];
      delete vdofs[i];
      delete el_x[i];
   }
}
 
Operator &BlockNonlinearForm::GetGradient(const Vector &x) const
{
   BlockVector bx(const_cast<Vector&>(x), block_trueOffsets);
   const BlockVector &pbx = Prolongate(bx);
 
   ComputeGradientBlocked(pbx);
 
   Array2D<SparseMatrix *> mGrads(fes.Size(), fes.Size());
   mGrads = Grads;
   if (needs_prolongation)
   {
      for (int s1 = 0; s1 < fes.Size(); ++s1)
      {
         for (int s2 = 0; s2 < fes.Size(); ++s2)
         {
            delete cGrads(s1, s2);
            cGrads(s1, s2) = RAP(*cP[s1], *Grads(s1, s2), *cP[s2]);
            mGrads(s1, s2) = cGrads(s1, s2);
         }
      }
   }
 
   for (int s = 0; s < fes.Size(); ++s)
   {
      for (int i = 0; i < ess_tdofs[s]->Size(); ++i)
      {
         for (int j = 0; j < fes.Size(); ++j)
         {
            if (s == j)
            {
               mGrads(s, s)->EliminateRowCol((*ess_tdofs[s])[i],
                                             Matrix::DIAG_ONE);
            }
            else
            {
               mGrads(s, j)->EliminateRow((*ess_tdofs[s])[i]);
               mGrads(j, s)->EliminateCol((*ess_tdofs[s])[i]);
            }
         }
      }
   }
 
   delete BlockGrad;
   BlockGrad = new BlockOperator(block_trueOffsets);
   for (int i = 0; i < fes.Size(); ++i)
   {
      for (int j = 0; j < fes.Size(); ++j)
      {
         BlockGrad->SetBlock(i, j, mGrads(i, j));
      }
   }
   return *BlockGrad;
}
 
BlockNonlinearForm::~BlockNonlinearForm()
{
   delete BlockGrad;
   for (int i=0; i<fes.Size(); ++i)
   {
      for (int j=0; j<fes.Size(); ++j)
      {
         delete Grads(i,j);
         delete cGrads(i,j);
      }
      delete ess_tdofs[i];
   }
 
   for (int i = 0; i < dnfi.Size(); ++i)
   {
      delete dnfi[i];
   }
 
   for (int i = 0; i < bnfi.Size(); ++i)
   {
      delete bnfi[i];
   }
 
   for (int i = 0; i < fnfi.Size(); ++i)
   {
      delete fnfi[i];
   }
 
   for (int i = 0; i < bfnfi.Size(); ++i)
   {
      delete bfnfi[i];
   }
 
}
 
}