PetscSolverFeti.cc 48.2 KB
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//
// Software License for AMDiS
//
// Copyright (c) 2010 Dresden University of Technology 
// All rights reserved.
// Authors: Simon Vey, Thomas Witkowski et al.
//
// This file is part of AMDiS
//
// See also license.opensource.txt in the distribution.


#include "parallel/PetscSolverFeti.h"
#include "parallel/StdMpi.h"
#include "parallel/MpiHelper.h"

namespace AMDiS {

  using namespace std;

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  FetiStatisticsData PetscSolverFeti::fetiStatistics;
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#ifdef HAVE_PETSC_DEV 
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  // y = mat * x
  int petscMultMatSchurPrimal(Mat mat, Vec x, Vec y)
  {
    // S_PiPi = K_PiPi - K_PiB inv(K_BB) K_BPi

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    double first = MPI::Wtime();
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    void *ctx;
    MatShellGetContext(mat, &ctx);
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    SchurPrimalData* data = static_cast<SchurPrimalData*>(ctx);
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    MatMult(*(data->mat_b_primal), x, data->tmp_vec_b);
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    data->fetiSolver->solveLocalProblem(data->tmp_vec_b, data->tmp_vec_b);
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    MatMult(*(data->mat_primal_b), data->tmp_vec_b, data->tmp_vec_primal);
    MatMult(*(data->mat_primal_primal), x, y);
    VecAXPBY(y, -1.0, 1.0, data->tmp_vec_primal);

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    PetscSolverFeti::fetiStatistics.nSchurPrimalApply++;
    PetscSolverFeti::fetiStatistics.timeSchurPrimalApply += MPI::Wtime() - first;

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    return 0;
  }


  // y = mat * x
  int petscMultMatFeti(Mat mat, Vec x, Vec y)
  {
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    //    F = L inv(K_BB) trans(L) + L inv(K_BB) K_BPi inv(S_PiPi) K_PiB inv(K_BB) trans(L)
    // => F = L [I + inv(K_BB) K_BPi inv(S_PiPi) K_PiB] inv(K_BB) trans(L)
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    void *ctx;
    MatShellGetContext(mat, &ctx);
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    FetiData* data = static_cast<FetiData*>(ctx);
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    // tmp_vec_b0 = inv(K_BB) trans(L) x
    MatMultTranspose(*(data->mat_lagrange), x, data->tmp_vec_b0);
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    data->fetiSolver->solveLocalProblem(data->tmp_vec_b0, data->tmp_vec_b0);
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    // tmp_vec_b1 = inv(K_BB) K_BPi  inv(S_PiPi) K_PiB tmp_vec_b0
    MatMult(*(data->mat_primal_b), data->tmp_vec_b0, data->tmp_vec_primal);
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    double first = MPI::Wtime();
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    KSPSolve(*(data->ksp_schur_primal), data->tmp_vec_primal, data->tmp_vec_primal);
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    PetscSolverFeti::fetiStatistics.nSchurPrimalSolve++;
    PetscSolverFeti::fetiStatistics.timeSchurPrimalSolve +=
      MPI::Wtime() - first;

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    MatMult(*(data->mat_b_primal), data->tmp_vec_primal, data->tmp_vec_b1);
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    // y = L (tmp_vec_b0 + tmp_vec_b1)
    VecAXPBY(data->tmp_vec_b0, 1.0, 1.0, data->tmp_vec_b1);
    MatMult(*(data->mat_lagrange), data->tmp_vec_b0, y);
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    PetscSolverFeti::fetiStatistics.nFetiApply++;

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    return 0;
  }


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  // y = PC * x
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  PetscErrorCode petscApplyFetiDirichletPrecon(PC pc, Vec x, Vec y)
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  {
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    // Get data for the preconditioner
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    void *ctx;
    PCShellGetContext(pc, &ctx);
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    FetiDirichletPreconData* data = static_cast<FetiDirichletPreconData*>(ctx);
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    // Multiply with scaled Lagrange constraint matrix.
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    MatMultTranspose(*(data->mat_lagrange_scaled), x, data->tmp_vec_b);


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    // === Restriction of the B nodes to the boundary nodes. ===
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    int nLocalB;
    int nLocalDuals;
    VecGetLocalSize(data->tmp_vec_b, &nLocalB);
    VecGetLocalSize(data->tmp_vec_duals0, &nLocalDuals);
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    PetscScalar *local_b, *local_duals;
    VecGetArray(data->tmp_vec_b, &local_b);
    VecGetArray(data->tmp_vec_duals0, &local_duals);

    for (int i = nLocalB - nLocalDuals, j = 0; i < nLocalB; i++, j++)
      local_duals[j] = local_b[i];
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    VecRestoreArray(data->tmp_vec_b, &local_b);
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    VecRestoreArray(data->tmp_vec_duals0, &local_duals);
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    // === K_DD - K_DI inv(K_II) K_ID ===
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    MatMult(*(data->mat_duals_duals), data->tmp_vec_duals0, data->tmp_vec_duals1);
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    MatMult(*(data->mat_interior_duals), data->tmp_vec_duals0, data->tmp_vec_interior);
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    KSPSolve(*(data->ksp_interior), data->tmp_vec_interior, data->tmp_vec_interior);
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    MatMult(*(data->mat_duals_interior), data->tmp_vec_interior, data->tmp_vec_duals0);

    VecAXPBY(data->tmp_vec_duals0, 1.0, -1.0, data->tmp_vec_duals1);


    // === Prolongation from local dual nodes to B nodes.

    VecGetArray(data->tmp_vec_b, &local_b);
    VecGetArray(data->tmp_vec_duals0, &local_duals);

    for (int i = nLocalB - nLocalDuals, j = 0; i < nLocalB; i++, j++)
      local_b[i] = local_duals[j];

    VecRestoreArray(data->tmp_vec_b, &local_b);
    VecRestoreArray(data->tmp_vec_duals0, &local_duals);


    // Multiply with scaled Lagrange constraint matrix.
    MatMult(*(data->mat_lagrange_scaled), data->tmp_vec_b, y);

    return 0;
  }


  // y = PC * x
  PetscErrorCode petscApplyFetiLumpedPrecon(PC pc, Vec x, Vec y)
  {
    // Get data for the preconditioner
    void *ctx;
    PCShellGetContext(pc, &ctx);
    FetiLumpedPreconData* data = static_cast<FetiLumpedPreconData*>(ctx);

    // Multiply with scaled Lagrange constraint matrix.
    MatMultTranspose(*(data->mat_lagrange_scaled), x, data->tmp_vec_b);
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    // === Restriction of the B nodes to the boundary nodes. ===
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    int nLocalB;
    int nLocalDuals;
    VecGetLocalSize(data->tmp_vec_b, &nLocalB);
    VecGetLocalSize(data->tmp_vec_duals0, &nLocalDuals);
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    PetscScalar *local_b, *local_duals;
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    VecGetArray(data->tmp_vec_b, &local_b);
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    VecGetArray(data->tmp_vec_duals0, &local_duals);
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    for (int i = nLocalB - nLocalDuals, j = 0; i < nLocalB; i++, j++)
      local_duals[j] = local_b[i];
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    VecRestoreArray(data->tmp_vec_b, &local_b);
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    VecRestoreArray(data->tmp_vec_duals0, &local_duals);


    // === K_DD ===

    MatMult(*(data->mat_duals_duals), data->tmp_vec_duals0, data->tmp_vec_duals1);

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    // === Prolongation from local dual nodes to B nodes.
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    VecGetArray(data->tmp_vec_b, &local_b);
    VecGetArray(data->tmp_vec_duals1, &local_duals);

    for (int i = nLocalB - nLocalDuals, j = 0; i < nLocalB; i++, j++)
      local_b[i] = local_duals[j];
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    VecRestoreArray(data->tmp_vec_b, &local_b);
    VecRestoreArray(data->tmp_vec_duals0, &local_duals);
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    // Multiply with scaled Lagrange constraint matrix.
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    MatMult(*(data->mat_lagrange_scaled), data->tmp_vec_b, y);

    return 0;
  }


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  PetscSolverFeti::PetscSolverFeti()
    : PetscSolver(),
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      nComponents(-1),
      schurPrimalSolver(0)
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  {
    FUNCNAME("PetscSolverFeti::PetscSolverFeti()");

    string preconditionerName = "";
    Parameters::get("parallel->solver->precon", preconditionerName);
    if (preconditionerName == "" || preconditionerName == "none") {
      fetiPreconditioner = FETI_NONE;
    } else if (preconditionerName == "dirichlet") {
      fetiPreconditioner = FETI_DIRICHLET;
    } else if (preconditionerName == "lumped") {
      fetiPreconditioner = FETI_LUMPED;
    } else {
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      ERROR_EXIT("Preconditioner \"%s\" not available!\n", 
		 preconditionerName.c_str());
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    }
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    Parameters::get("parallel->feti->schur primal solver", schurPrimalSolver);
    TEST_EXIT(schurPrimalSolver == 0 || schurPrimalSolver == 1)
      ("Wrong solver \"%d\"for the Schur primal complement!\n", 
       schurPrimalSolver);
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  }


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  void PetscSolverFeti::updateDofData()
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  {
    FUNCNAME("PetscSolverFeti::updateDofData()");
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    TEST_EXIT(meshDistributor->getFeSpace()->getBasisFcts()->getDegree() == 1)
      ("Works for linear basis functions only!\n");
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    createPrimals();

    createDuals();

    createLagrange();

    createIndexB();
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  }


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  void PetscSolverFeti::createPrimals()
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  {
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    FUNCNAME("PetscSolverFeti::createPrimals()");  
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    // === Define all vertices on the interior boundaries of the macro mesh ===
    // === to be primal variables.                                          ===

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    primals.clear();
    DofContainerSet& vertices = 
      meshDistributor->getBoundaryDofInfo().geoDofs[VERTEX];
    TEST_EXIT_DBG(vertices.size())("No primal vertices on this rank!\n");
    for (DofContainerSet::iterator it = vertices.begin(); 
	 it != vertices.end(); ++it)
      primals.insert(**it);
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    // === Calculate the number of primals that are owned by the rank and ===
    // === create local indices of the primals starting at zero.          ===

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    globalPrimalIndex.clear();
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    nRankPrimals = 0;
    for (DofIndexSet::iterator it = primals.begin(); it != primals.end(); ++it)
      if (meshDistributor->getIsRankDof(*it)) {
	globalPrimalIndex[*it] = nRankPrimals;
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	nRankPrimals++;
      }

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    // === Get overall number of primals and rank's displacement in the ===
    // === numbering of the primals.                                    ===

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    nOverallPrimals = 0;
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    rStartPrimals = 0;
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    mpi::getDofNumbering(meshDistributor->getMpiComm(),
			 nRankPrimals, rStartPrimals, nOverallPrimals);

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    // === Create global primal index for all primals. ===

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    for (DofMapping::iterator it = globalPrimalIndex.begin();
	 it != globalPrimalIndex.end(); ++it)
      it->second += rStartPrimals;

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    MSG("nRankPrimals = %d   nOverallPrimals = %d\n", 
	nRankPrimals, nOverallPrimals);
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    // === Communicate primal's global index from ranks that own the     ===
    // === primals to ranks that contain this primals but are not owning ===
    // === them.                                                         ===

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    StdMpi<vector<int> > stdMpi(meshDistributor->getMpiComm());
    RankToDofContainer& sendDofs = meshDistributor->getSendDofs();
    for (RankToDofContainer::iterator it = sendDofs.begin();
	 it != sendDofs.end(); ++it)
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)
	if (globalPrimalIndex.count(**dofIt))
	  stdMpi.getSendData(it->first).push_back(globalPrimalIndex[**dofIt]);
    stdMpi.updateSendDataSize();

    RankToDofContainer& recvDofs = meshDistributor->getRecvDofs();
    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      bool recvFromRank = false;
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)
	if (primals.count(**dofIt) && 
	    meshDistributor->getIsRankDof(**dofIt) == false) {
	  recvFromRank = true;
	  break;
	}

      if (recvFromRank) 
	stdMpi.recv(it->first);
    }
    stdMpi.startCommunication();

    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      int i = 0;
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt) {
	if (primals.count(**dofIt) && 
	    meshDistributor->getIsRankDof(**dofIt) == false)
	  globalPrimalIndex[**dofIt] = stdMpi.getRecvData(it->first)[i++];
      }
    }

    TEST_EXIT_DBG(primals.size() == globalPrimalIndex.size())
      ("Number of primals %d, but number of global primals on this rank is %d!\n",
       primals.size(), globalPrimalIndex.size());


    TEST_EXIT_DBG(nOverallPrimals > 0)
      ("There are zero primal nodes in domain!\n");
  }


  void PetscSolverFeti::createDuals()
  {
    FUNCNAME("PetscSolverFeti::createDuals()");
    
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    // === Create for each dual node that is owned by the rank, the set ===
    // === of ranks that contain this node (denoted by W(x_j)).         ===
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    boundaryDofRanks.clear();

    RankToDofContainer& sendDofs = meshDistributor->getSendDofs();
    for (RankToDofContainer::iterator it = sendDofs.begin();
	 it != sendDofs.end(); ++it) {
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt) {
	// If DOF is not primal, i.e., its a dual node
	if (primals.count(**dofIt) == 0) {
	  boundaryDofRanks[**dofIt].insert(mpiRank);
	  boundaryDofRanks[**dofIt].insert(it->first);
	}
      }
    }

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    // === Communicate these sets for all rank owned dual nodes to other ===
    // === ranks that also have this node.                               ===

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    StdMpi<vector<std::set<int> > > stdMpi(meshDistributor->getMpiComm());
    for (RankToDofContainer::iterator it = sendDofs.begin();
	 it != sendDofs.end(); ++it)
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)
	if (primals.count(**dofIt) == 0)
	  stdMpi.getSendData(it->first).push_back(boundaryDofRanks[**dofIt]);

    stdMpi.updateSendDataSize();

    RankToDofContainer& recvDofs = meshDistributor->getRecvDofs();
    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      bool recvFromRank = false;
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)
	if (primals.count(**dofIt) == 0) {
	  recvFromRank = true;
	  break;
	}

      if (recvFromRank)
	stdMpi.recv(it->first);
    }
    stdMpi.startCommunication();

    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      int i = 0;
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)	
	if (primals.count(**dofIt) == 0)
	  boundaryDofRanks[**dofIt] = stdMpi.getRecvData(it->first)[i++];	      
    }


    // === Create global index of the dual nodes on each rank. ===

    duals.clear();
    globalDualIndex.clear();

    int nRankAllDofs = meshDistributor->getFeSpace()->getAdmin()->getUsedDofs();
    nRankB = nRankAllDofs - primals.size();
    nOverallB = 0;
    rStartB = 0;
    mpi::getDofNumbering(meshDistributor->getMpiComm(),
			 nRankB, rStartB, nOverallB);
    DofContainer allBoundaryDofs;
    meshDistributor->getAllBoundaryDofs(allBoundaryDofs);
    int nRankInteriorDofs = nRankAllDofs - allBoundaryDofs.size();

    int nRankDuals = 0;
    for (DofContainer::iterator it = allBoundaryDofs.begin();
	 it != allBoundaryDofs.end(); ++it) {
      if (primals.count(**it) == 0) {
	duals.insert(**it);
	globalDualIndex[**it] = rStartB + nRankInteriorDofs + nRankDuals;
	nRankDuals++;
      }
    }

    int nOverallDuals = nRankDuals;
    mpi::globalAdd(nOverallDuals);

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    MSG("nRankDuals = %d   nOverallDuals = %d\n",
	nRankDuals, nOverallDuals);
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  }

  
  void PetscSolverFeti::createLagrange()
  {
    FUNCNAME("PetscSolverFeti::createLagrange()");

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    // === Reserve for each dual node, on the rank that owns this node, the ===
    // === appropriate number of Lagrange constraints.                      ===

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    nRankLagrange = 0;
    for (DofIndexSet::iterator it = duals.begin(); it != duals.end(); ++it) {
      if (meshDistributor->getIsRankDof(*it)) {
	dofFirstLagrange[*it] = nRankLagrange;
	int degree = boundaryDofRanks[*it].size();
	nRankLagrange += (degree * (degree - 1)) / 2;
      }
    }

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    // === Get the overall number of Lagrange constraints and create the ===
    // === mapping dofFirstLagrange, that defines for each dual boundary ===
    // === node the first Lagrange constraint global index.              ===

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    nOverallLagrange = 0;
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    rStartLagrange = 0;
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    mpi::getDofNumbering(meshDistributor->getMpiComm(),
			 nRankLagrange, rStartLagrange, nOverallLagrange);

    for (DofIndexSet::iterator it = duals.begin(); it != duals.end(); ++it)
      if (meshDistributor->getIsRankDof(*it))
	dofFirstLagrange[*it] += rStartLagrange;

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    MSG("nRankLagrange = %d  nOverallLagrange = %d\n",
	nRankLagrange, nOverallLagrange);
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    // === Communicate dofFirstLagrange to all other ranks. ===
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    StdMpi<vector<int> > stdMpi(meshDistributor->getMpiComm());
    RankToDofContainer& sendDofs = meshDistributor->getSendDofs();
    for (RankToDofContainer::iterator it = sendDofs.begin();
	 it != sendDofs.end(); ++it)
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt) {
	if (primals.count(**dofIt) == 0) {
	  TEST_EXIT_DBG(dofFirstLagrange.count(**dofIt))("Should not happen!\n");
	  stdMpi.getSendData(it->first).push_back(dofFirstLagrange[**dofIt]);
	}
      }
    stdMpi.updateSendDataSize();

    RankToDofContainer& recvDofs = meshDistributor->getRecvDofs();
    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      bool recvData = false;
      for (DofContainer::iterator dofIt = it->second.begin();
	   dofIt != it->second.end(); ++dofIt)
	if (primals.count(**dofIt) == 0) {
	  recvData = true;
	  break;
	}
	  
      if (recvData)
	stdMpi.recv(it->first);
    }

    stdMpi.startCommunication();

    for (RankToDofContainer::iterator it = recvDofs.begin();
	 it != recvDofs.end(); ++it) {
      int counter = 0;
      for (unsigned int i = 0; i < it->second.size(); i++)
	if (primals.count(*(it->second[i])) == 0)
	  dofFirstLagrange[*(it->second[i])] = stdMpi.getRecvData(it->first)[counter++];
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    }     
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  }


  void PetscSolverFeti::createIndexB()
  {
    FUNCNAME("PetscSolverFeti::createIndeB()");

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    localIndexB.clear();
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    DOFAdmin* admin = meshDistributor->getFeSpace()->getAdmin();
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    // === To ensure that all interior node on each rank are listen first in ===
    // === the global index of all B nodes, insert all interior nodes first, ===
    // === without defining a correct index.                                 ===
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    for (int i = 0; i < admin->getUsedSize(); i++)
      if (admin->isDofFree(i) == false && primals.count(i) == 0)
	if (duals.count(i) == 0 && primals.count(i) == 0)
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	  localIndexB[i] = -1;
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    // === Get correct index for all interior nodes. ===

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    nLocalInterior = 0;
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    for (DofMapping::iterator it = localIndexB.begin(); 
	 it != localIndexB.end(); ++it) {
      it->second = nLocalInterior;
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      nLocalInterior++;
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    }
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    nLocalDuals = duals.size();
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    TEST_EXIT_DBG(nLocalInterior + primals.size() + duals.size() == 
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		  static_cast<unsigned int>(admin->getUsedDofs()))
      ("Should not happen!\n");

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    // === And finally, add the global indicies of all dual nodes. ===

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    for (DofIndexSet::iterator it = duals.begin();
	 it != duals.end(); ++it)
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      localIndexB[*it] = globalDualIndex[*it] - rStartB;
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  }


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  void PetscSolverFeti::createMatLagrange()
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  {
    FUNCNAME("PetscSolverFeti::createMatLagrange()");

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    // === Create distributed matrix for Lagrange constraints. ===

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    MatCreateMPIAIJ(PETSC_COMM_WORLD,
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		    nRankLagrange * nComponents, 
		    nRankB * nComponents,
		    nOverallLagrange * nComponents, 
		    nOverallB * nComponents,
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		    2, PETSC_NULL, 2, PETSC_NULL,
		    &mat_lagrange);

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    // === Create for all duals the corresponding Lagrange constraints. On ===
    // === each rank we traverse all pairs (n, m) of ranks, with n < m,    ===
    // === that contain this node. If the current rank number is r, and    ===
    // === n == r, the rank sets 1.0 for the corresponding constraint, if  ===
    // === m == r, than the rank sets -1.0 for the corresponding           ===
    // === constraint.                                                     ===

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    for (DofIndexSet::iterator it = duals.begin(); it != duals.end(); ++it) {
      TEST_EXIT_DBG(dofFirstLagrange.count(*it))("Should not happen!\n");
      TEST_EXIT_DBG(boundaryDofRanks.count(*it))("Should not happen!\n");

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      // Global index of the first Lagrange constriant for this node.
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      int index = dofFirstLagrange[*it];
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      // Copy set of all ranks that contain this dual node.
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      vector<int> W(boundaryDofRanks[*it].begin(), boundaryDofRanks[*it].end());
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      // Number of ranks that contain this dual node.
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      int degree = W.size();

      TEST_EXIT_DBG(globalDualIndex.count(*it))("Should not happen!\n");
      int dualCol = globalDualIndex[*it];

      for (int i = 0; i < degree; i++) {
	for (int j = i + 1; j < degree; j++) {
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	  if (W[i] == mpiRank || W[j] == mpiRank) {
	    // Set the constraint for all components of the system.
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	    for (int k = 0; k < nComponents; k++) {
	      int rowIndex = index * nComponents + k;
	      int colIndex = dualCol * nComponents + k;
	      double value = (W[i] == mpiRank ? 1.0 : -1.0);
	      MatSetValue(mat_lagrange, rowIndex, colIndex, value, 
			  INSERT_VALUES);
	    }
	  }

	  index++;
	}
      }
    }

    MatAssemblyBegin(mat_lagrange, MAT_FINAL_ASSEMBLY);
    MatAssemblyEnd(mat_lagrange, MAT_FINAL_ASSEMBLY);
  }


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  void PetscSolverFeti::createSchurPrimalKsp()
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  {
    FUNCNAME("PetscSolverFeti::createSchurPrimal()");

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    if (schurPrimalSolver == 0) {
      schurPrimalData.mat_primal_primal = &mat_primal_primal;
      schurPrimalData.mat_primal_b = &mat_primal_b;
      schurPrimalData.mat_b_primal = &mat_b_primal;
      schurPrimalData.fetiSolver = this;
      
      VecCreateMPI(PETSC_COMM_WORLD, 
		   nRankB * nComponents, nOverallB * nComponents,
		   &(schurPrimalData.tmp_vec_b));
      VecCreateMPI(PETSC_COMM_WORLD, 
		   nRankPrimals * nComponents, nOverallPrimals * nComponents,
		   &(schurPrimalData.tmp_vec_primal));

      MatCreateShell(PETSC_COMM_WORLD,
		     nRankPrimals * nComponents, nRankPrimals * nComponents,
		     nOverallPrimals * nComponents, nOverallPrimals * nComponents,
		     &schurPrimalData, 
		     &mat_schur_primal);
      MatShellSetOperation(mat_schur_primal, MATOP_MULT, 
			   (void(*)(void))petscMultMatSchurPrimal);
      
      KSPCreate(PETSC_COMM_WORLD, &ksp_schur_primal);
      KSPSetOperators(ksp_schur_primal, mat_schur_primal, mat_schur_primal, SAME_NONZERO_PATTERN);
      KSPSetOptionsPrefix(ksp_schur_primal, "solver_sp_");
      KSPSetFromOptions(ksp_schur_primal);
    } else {
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      TEST_EXIT(ksp_b)("No solver object for local problem!\n");

      double wtime = MPI::Wtime();

      Vec tmpVecB;
      VecCreateMPI(PETSC_COMM_WORLD,
		   nRankB * nComponents,
		   nOverallB * nComponents, &tmpVecB);

      int nRowsRankPrimal = nRankPrimals * nComponents;
      int nRowsOverallPrimal = nOverallPrimals * nComponents;
      Vec tmpVecPrimal;
      VecCreateMPI(PETSC_COMM_WORLD,
		   nRowsRankPrimal, nRowsOverallPrimal, &tmpVecPrimal);
      Mat matPrimal;
      MatCreateMPIAIJ(PETSC_COMM_WORLD,
		      nRowsRankPrimal, nRowsRankPrimal, 
		      nRowsOverallPrimal, nRowsOverallPrimal,
		      10, PETSC_NULL, 10, PETSC_NULL, &matPrimal);
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      PetscInt nCols;
      const PetscInt *cols;
      const PetscScalar *values;
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      int nLocalPrimals = globalPrimalIndex.size() * nComponents;
      map<int, vector<pair<int, double> > > mat_b_primal_cols;

      for (int i = 0; i < nRankB * nComponents; i++) {
	PetscInt row = rStartB * nComponents + i;
	MatGetRow(mat_b_primal, row, &nCols, &cols, &values);

	for (int j = 0; j < nCols; j++)
	  if (values[j] != 0.0)
	    mat_b_primal_cols[cols[j]].push_back(make_pair(i, values[j]));

	MatRestoreRow(mat_b_primal, i, &nCols, &cols, &values);
      }

      int maxLocalPrimal = mat_b_primal_cols.size();
      mpi::globalMax(maxLocalPrimal);

      TEST_EXIT(mat_b_primal_cols.size() ==
		globalPrimalIndex.size() * nComponents)("Should not happen!\n");
      for (map<int, vector<pair<int, double> > >::iterator it = mat_b_primal_cols.begin();
	   it != mat_b_primal_cols.end(); ++it) {
	Vec tmpVec;
	VecCreateSeq(PETSC_COMM_SELF, nRankB * nComponents, &tmpVec);

 	for (unsigned int i = 0; i < it->second.size(); i++) 
 	  VecSetValue(tmpVec, 
 		      it->second[i].first, it->second[i].second, INSERT_VALUES);

	VecAssemblyBegin(tmpVec);
	VecAssemblyEnd(tmpVec);

       	KSPSolve(ksp_b, tmpVec, tmpVec);

	PetscScalar *tmpValuesB, *tmpValues;
	VecGetArray(tmpVecB, &tmpValuesB);
	VecGetArray(tmpVec, &tmpValues);
	
	for (int i  = 0; i < nRankB * nComponents; i++)
	  tmpValuesB[i] = tmpValues[i];
	
	VecRestoreArray(tmpVec, &tmpValues);
	VecRestoreArray(tmpVecB, &tmpValuesB);

	MatMult(mat_primal_b, tmpVecB, tmpVecPrimal);

	VecGetArray(tmpVecPrimal, &tmpValues);

	for (int i = 0; i < nRowsRankPrimal; i++)
	    MatSetValue(matPrimal, rStartPrimals * nComponents + i, 
			it->first, tmpValues[i], ADD_VALUES);

	VecRestoreArray(tmpVecPrimal, &tmpValues);
	VecDestroy(&tmpVec);
      }

      for (int i = mat_b_primal_cols.size(); i < maxLocalPrimal; i++) 
	MatMult(mat_primal_b, tmpVecB, tmpVecPrimal);

      VecDestroy(&tmpVecB);
      VecDestroy(&tmpVecPrimal);

      MatAssemblyBegin(matPrimal, MAT_FINAL_ASSEMBLY);
      MatAssemblyEnd(matPrimal, MAT_FINAL_ASSEMBLY);

      MatAXPY(mat_primal_primal, -1.0, matPrimal, DIFFERENT_NONZERO_PATTERN);

      MatDestroy(&matPrimal);

      MatInfo minfo;
      MatGetInfo(mat_primal_primal, MAT_GLOBAL_SUM, &minfo);
      MSG("Schur primal matrix nnz = %f\n", minfo.nz_used);

      KSPCreate(PETSC_COMM_WORLD, &ksp_schur_primal);
      KSPSetOperators(ksp_schur_primal, mat_primal_primal, 
		      mat_primal_primal, SAME_NONZERO_PATTERN);
      KSPSetOptionsPrefix(ksp_schur_primal, "solver_sp_");
      KSPSetFromOptions(ksp_schur_primal);
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      MSG("Creating Schur primal matrix needed %.5f seconds.\n",
	  MPI::Wtime() - wtime);
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    }
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  }


  void PetscSolverFeti::destroySchurPrimalKsp()
  {
    FUNCNAME("PetscSolverFeti::destroySchurPrimal()");

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    if (schurPrimalSolver == 0) {
      schurPrimalData.mat_primal_primal = PETSC_NULL;
      schurPrimalData.mat_primal_b = PETSC_NULL;
      schurPrimalData.mat_b_primal = PETSC_NULL;
      schurPrimalData.fetiSolver = NULL;
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      VecDestroy(&schurPrimalData.tmp_vec_b);
      VecDestroy(&schurPrimalData.tmp_vec_primal);
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      MatDestroy(&mat_schur_primal);
      KSPDestroy(&ksp_schur_primal);
    } else {
      KSPDestroy(&ksp_schur_primal);
    }
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  }


  void PetscSolverFeti::createFetiKsp()
  {
    FUNCNAME("PetscSolverFeti::createFetiKsp()");

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    // === Create FETI-DP solver object. ===

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    fetiData.mat_primal_b = &mat_primal_b;
    fetiData.mat_b_primal = &mat_b_primal;
    fetiData.mat_lagrange = &mat_lagrange;
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    fetiData.fetiSolver = this;
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    fetiData.ksp_schur_primal = &ksp_schur_primal;
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    VecCreateMPI(PETSC_COMM_WORLD, 
		 nRankB * nComponents, nOverallB * nComponents,
		 &(fetiData.tmp_vec_b0));
    VecCreateMPI(PETSC_COMM_WORLD, 
		 nRankB * nComponents, nOverallB * nComponents,
		 &(fetiData.tmp_vec_b1));
    VecCreateMPI(PETSC_COMM_WORLD, 
		 nRankPrimals * nComponents, nOverallPrimals * nComponents,
		 &(fetiData.tmp_vec_primal));
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    MatCreateShell(PETSC_COMM_WORLD,
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		   nRankLagrange * nComponents, nRankLagrange * nComponents,
		   nOverallLagrange * nComponents, nOverallLagrange * nComponents,
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		   &fetiData, &mat_feti);
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    MatShellSetOperation(mat_feti, MATOP_MULT, (void(*)(void))petscMultMatFeti);


    KSPCreate(PETSC_COMM_WORLD, &ksp_feti);
    KSPSetOperators(ksp_feti, mat_feti, mat_feti, SAME_NONZERO_PATTERN);
    KSPSetOptionsPrefix(ksp_feti, "solver_feti_");
    KSPSetFromOptions(ksp_feti);
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    // === Create FETI-DP preconditioner object. ===
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    if (fetiPreconditioner != FETI_NONE) {
      MatDuplicate(mat_lagrange, MAT_COPY_VALUES, &mat_lagrange_scaled);
      MatScale(mat_lagrange_scaled, 0.5);
    }
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    switch (fetiPreconditioner) {
    case FETI_DIRICHLET:           
      KSPCreate(PETSC_COMM_SELF, &ksp_interior);
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      KSPSetOperators(ksp_interior, mat_interior_interior, mat_interior_interior, 
		      SAME_NONZERO_PATTERN);
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      KSPSetOptionsPrefix(ksp_interior, "solver_interior_");
      KSPSetFromOptions(ksp_interior);
            
      fetiDirichletPreconData.mat_lagrange_scaled = &mat_lagrange_scaled;
      fetiDirichletPreconData.mat_interior_interior = &mat_interior_interior;
      fetiDirichletPreconData.mat_duals_duals = &mat_duals_duals;
      fetiDirichletPreconData.mat_interior_duals = &mat_interior_duals;
      fetiDirichletPreconData.mat_duals_interior = &mat_duals_interior;
      fetiDirichletPreconData.ksp_interior = &ksp_interior;
      
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      VecCreateMPI(PETSC_COMM_WORLD, 
		   nRankB * nComponents, nOverallB * nComponents,
		   &(fetiDirichletPreconData.tmp_vec_b));      
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      MatGetVecs(mat_duals_duals, PETSC_NULL, &(fetiDirichletPreconData.tmp_vec_duals0));
      MatGetVecs(mat_duals_duals, PETSC_NULL, &(fetiDirichletPreconData.tmp_vec_duals1));
      MatGetVecs(mat_interior_interior, PETSC_NULL, &(fetiDirichletPreconData.tmp_vec_interior));
      
      KSPGetPC(ksp_feti, &precon_feti);
      PCSetType(precon_feti, PCSHELL);
      PCShellSetContext(precon_feti, static_cast<void*>(&fetiDirichletPreconData));
      PCShellSetApply(precon_feti, petscApplyFetiDirichletPrecon);
      
      break;

    case FETI_LUMPED:
      fetiLumpedPreconData.mat_lagrange_scaled = &mat_lagrange_scaled;
      fetiLumpedPreconData.mat_duals_duals = &mat_duals_duals;

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      VecCreateMPI(PETSC_COMM_WORLD, 
		   nRankB * nComponents, nOverallB * nComponents,
		   &(fetiLumpedPreconData.tmp_vec_b));
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      MatGetVecs(mat_duals_duals, PETSC_NULL, &(fetiLumpedPreconData.tmp_vec_duals0));
      MatGetVecs(mat_duals_duals, PETSC_NULL, &(fetiLumpedPreconData.tmp_vec_duals1));

      KSPGetPC(ksp_feti, &precon_feti);
      PCSetType(precon_feti, PCSHELL);
      PCShellSetContext(precon_feti, static_cast<void*>(&fetiLumpedPreconData));
      PCShellSetApply(precon_feti, petscApplyFetiLumpedPrecon);
      
      break;
    }
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  }
  

  void PetscSolverFeti::destroyFetiKsp()
  {
    FUNCNAME("PetscSolverFeti::destroyFetiKsp()");

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    // === Destroy FETI-DP solver object. ===

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    fetiData.mat_primal_b = PETSC_NULL;
    fetiData.mat_b_primal = PETSC_NULL;
    fetiData.mat_lagrange = PETSC_NULL;
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    fetiData.fetiSolver = NULL;
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    fetiData.ksp_schur_primal = PETSC_NULL;
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    VecDestroy(&fetiData.tmp_vec_b0);
    VecDestroy(&fetiData.tmp_vec_b1);
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    VecDestroy(&fetiData.tmp_vec_primal);
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    MatDestroy(&mat_feti);
    KSPDestroy(&ksp_feti);
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    // === Destroy FETI-DP preconditioner object. ===

    switch (fetiPreconditioner) {
    case FETI_DIRICHLET:           
      KSPDestroy(&ksp_interior);

      fetiDirichletPreconData.mat_lagrange_scaled = NULL;
      fetiDirichletPreconData.mat_interior_interior = NULL;
      fetiDirichletPreconData.mat_duals_duals = NULL;
      fetiDirichletPreconData.mat_interior_duals = NULL;
      fetiDirichletPreconData.mat_duals_interior = NULL;
      fetiDirichletPreconData.ksp_interior = NULL;
      
      VecDestroy(&fetiDirichletPreconData.tmp_vec_b);
      VecDestroy(&fetiDirichletPreconData.tmp_vec_duals0);
      VecDestroy(&fetiDirichletPreconData.tmp_vec_duals1);
      VecDestroy(&fetiDirichletPreconData.tmp_vec_interior);
      MatDestroy(&mat_lagrange_scaled);
      break;

    case FETI_LUMPED:
      fetiLumpedPreconData.mat_lagrange_scaled = NULL;
      fetiLumpedPreconData.mat_duals_duals = NULL;

      VecDestroy(&fetiLumpedPreconData.tmp_vec_b);
      VecDestroy(&fetiLumpedPreconData.tmp_vec_duals0);
      VecDestroy(&fetiLumpedPreconData.tmp_vec_duals1);
      break;
    }
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  }


  void PetscSolverFeti::recoverSolution(Vec &vec_sol_b,
					Vec &vec_sol_primal,
					SystemVector &vec)
  {
    FUNCNAME("PetscSolverFeti::recoverSolution()");

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    // === Get local part of the solution for B variables. ===
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    PetscScalar *localSolB;
    VecGetArray(vec_sol_b, &localSolB);


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    // === Create scatter to get solutions of all primal nodes that are ===
    // === contained in rank's domain.                                  ===
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    vector<PetscInt> globalIsIndex, localIsIndex;
    globalIsIndex.reserve(globalPrimalIndex.size() * nComponents);
    localIsIndex.reserve(globalPrimalIndex.size() * nComponents);

    {
      int counter = 0;
      for (DofMapping::iterator it = globalPrimalIndex.begin();
	   it != globalPrimalIndex.end(); ++it) {
	for (int i = 0; i < nComponents; i++) {
	  globalIsIndex.push_back(it->second * nComponents + i);
	  localIsIndex.push_back(counter++);
	}
      }
    }
    
    IS globalIs, localIs;
    ISCreateGeneral(PETSC_COMM_SELF, 
		    globalIsIndex.size(), 
		    &(globalIsIndex[0]),
		    PETSC_USE_POINTER,
		    &globalIs);

    ISCreateGeneral(PETSC_COMM_SELF, 
		    localIsIndex.size(), 
		    &(localIsIndex[0]),
		    PETSC_USE_POINTER,
		    &localIs);

    Vec local_sol_primal;
    VecCreateSeq(PETSC_COMM_SELF, localIsIndex.size(), &local_sol_primal);

    VecScatter primalScatter;
    VecScatterCreate(vec_sol_primal, globalIs, local_sol_primal, localIs, &primalScatter);
    VecScatterBegin(primalScatter, vec_sol_primal, local_sol_primal, 
		    INSERT_VALUES, SCATTER_FORWARD);
    VecScatterEnd(primalScatter, vec_sol_primal, local_sol_primal, 
		  INSERT_VALUES, SCATTER_FORWARD);

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    ISDestroy(&globalIs);
    ISDestroy(&localIs);    
    VecScatterDestroy(&primalScatter);    
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    PetscScalar *localSolPrimal;
    VecGetArray(local_sol_primal, &localSolPrimal);


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    // === And copy from PETSc local vectors to the DOF vectors. ===
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    for (int i = 0; i < nComponents; i++) {
      DOFVector<double>& dofVec = *(vec.getDOFVector(i));

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      for (DofMapping::iterator it = localIndexB.begin();
	   it != localIndexB.end(); ++it) {
	int petscIndex = it->second * nComponents + i;
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	dofVec[it->first] = localSolB[petscIndex];
      }

      int counter = 0;
      for (DofMapping::iterator it = globalPrimalIndex.begin();
	   it != globalPrimalIndex.end(); ++it) {
	dofVec[it->first] = localSolPrimal[counter * nComponents + i];
	counter++;
      }
    }



    VecRestoreArray(vec_sol_b, &localSolB);
    VecRestoreArray(local_sol_primal, &localSolPrimal);
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    VecDestroy(&local_sol_primal);
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  }


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  void PetscSolverFeti::fillPetscMatrix(Matrix<DOFMatrix*> *mat)
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  {
    FUNCNAME("PetscSolverFeti::fillPetscMatrix()");   
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    // === Create all sets and indices. ===

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    updateDofData();

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    // === Create matrices for the FETI-DP method. ===

    int nRowsRankB = nRankB * nComponents;
    int nRowsOverallB = nOverallB * nComponents;
    int nRowsRankPrimal = nRankPrimals * nComponents;
    int nRowsOverallPrimal = nOverallPrimals * nComponents;
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    int nRowsInterior = nLocalInterior * nComponents;
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    int nRowsDual = nLocalDuals * nComponents;
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    MatCreateSeqAIJ(PETSC_COMM_SELF, nRowsRankB, nRowsRankB, 30, PETSC_NULL,
		    &mat_b_b);
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    MatCreateMPIAIJ(PETSC_COMM_WORLD,
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		    nRowsRankPrimal, nRowsRankPrimal, 
		    nRowsOverallPrimal, nRowsOverallPrimal,
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		    30, PETSC_NULL, 30, PETSC_NULL, &mat_primal_primal);
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    MatCreateMPIAIJ(PETSC_COMM_WORLD,
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		    nRowsRankB, nRowsRankPrimal, 
		    nRowsOverallB, nRowsOverallPrimal,
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		    30, PETSC_NULL, 30, PETSC_NULL, &mat_b_primal);
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    MatCreateMPIAIJ(PETSC_COMM_WORLD,
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		    nRowsRankPrimal, nRowsRankB,
		    nRowsOverallPrimal, nRowsOverallB,
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		    10, PETSC_NULL, 10, PETSC_NULL, &mat_primal_b);
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    // === Create matrices for FETI-DP preconditioner. ===

    if (fetiPreconditioner != FETI_NONE)
      MatCreateSeqAIJ(PETSC_COMM_SELF,
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		      nRowsDual, nRowsDual, 30, PETSC_NULL,
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		      &mat_duals_duals);

    if (fetiPreconditioner == FETI_DIRICHLET) {
      MatCreateSeqAIJ(PETSC_COMM_SELF,
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		      nRowsInterior, nRowsInterior, 30, PETSC_NULL,
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		      &mat_interior_interior);
      
      MatCreateSeqAIJ(PETSC_COMM_SELF,
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		      nRowsInterior, nRowsDual, 30, PETSC_NULL,
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		      &mat_interior_duals);
      
      MatCreateSeqAIJ(PETSC_COMM_SELF,
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		      nRowsDual, nRowsInterior, 30, PETSC_NULL,
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		      &mat_duals_interior);
    }
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    // === Prepare traverse of sequentially created matrices. ===
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    using mtl::tag::row; using mtl::tag::nz; using mtl::begin; using mtl::end;
    namespace traits = mtl::traits;
    typedef DOFMatrix::base_matrix_type Matrix;

    typedef traits::range_generator<row, Matrix>::type cursor_type;
    typedef traits::range_generator<nz, cursor_type>::type icursor_type;

    vector<int> cols, colsOther;
    vector<double> values, valuesOther;
    cols.reserve(300);
    colsOther.reserve(300);
    values.reserve(300);
    valuesOther.reserve(300);

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    vector<int> colsLocal, colsLocalOther;
    vector<double> valuesLocal, valuesLocalOther;
    colsLocal.reserve(300);
    colsLocalOther.reserve(300);
    valuesLocal.reserve(300);
    valuesLocalOther.reserve(300);

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    // === Traverse all sequentially created matrices and add the values to ===
    // === the global PETSc matrices.                                       ===

    for (int i = 0; i < nComponents; i++) {
      for (int j = 0; j < nComponents; j++) {
	if (!(*mat)[i][j])
	  continue;

	traits::col<Matrix>::type col((*mat)[i][j]->getBaseMatrix());
	traits::const_value<Matrix>::type value((*mat)[i][j]->getBaseMatrix());
	
	// Traverse all rows.
	for (cursor_type cursor = begin<row>((*mat)[i][j]->getBaseMatrix()), 
	       cend = end<row>((*mat)[i][j]->getBaseMatrix()); cursor != cend; ++cursor) {
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	  bool rowPrimal = primals.count(*cursor) != 0;
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	  cols.clear();
	  colsOther.clear();
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	  values.clear();	  
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	  valuesOther.clear();
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	  colsLocal.clear();
	  colsLocalOther.clear();
	  valuesLocal.clear();
	  valuesLocalOther.clear();

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	  // Traverse all columns.
	  for (icursor_type icursor = begin<nz>(cursor), icend = end<nz>(cursor); 
	       icursor != icend; ++icursor) {

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	    bool colPrimal = primals.count(col(*icursor)) != 0;

	    if (colPrimal) {
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	      // Column is a primal variable.

	      TEST_EXIT_DBG(globalPrimalIndex.count(col(*icursor)))
		("No global primal index for DOF %d!\n", col(*icursor));
	      
	      int colIndex = globalPrimalIndex[col(*icursor)] * nComponents + j;
	      
	      if (rowPrimal) {
		cols.push_back(colIndex);
		values.push_back(value(*icursor));
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	      } else {
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		colsOther.push_back(colIndex);
		valuesOther.push_back(value(*icursor));
	      }
	    } else {
	      // Column is not a primal variable.

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	      TEST_EXIT_DBG(localIndexB.count(col(*icursor)))
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		("No global B index for DOF %d!\n", col(*icursor));
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	      int colIndex = 
		(localIndexB[col(*icursor)] + rStartB) * nComponents + j;
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	      if (rowPrimal) {
		colsOther.push_back(colIndex);
		valuesOther.push_back(value(*icursor));
	      } else {
		cols.push_back(colIndex);
		values.push_back(value(*icursor));
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	      }
	    }
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	    // === For preconditioner ===

	    if (!rowPrimal && !colPrimal) {
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	      int rowIndex = localIndexB[*cursor];
	      int colIndex = localIndexB[col(*icursor)];
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	      if (rowIndex < nLocalInterior) {
		if (colIndex < nLocalInterior) {
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		  int colIndex2 = localIndexB[col(*icursor)] * nComponents + j;
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		  colsLocal.push_back(colIndex2);
		  valuesLocal.push_back(value(*icursor));
		} else {
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		  int colIndex2 = (localIndexB[col(*icursor)] - nLocalInterior) *
		    nComponents + j;
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		  colsLocalOther.push_back(colIndex2);
		  valuesLocalOther.push_back(value(*icursor));
		}
	      } else {
		if (colIndex < nLocalInterior) {
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		  int colIndex2 = localIndexB[col(*icursor)] * nComponents + j;
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		  colsLocalOther.push_back(colIndex2);
		  valuesLocalOther.push_back(value(*icursor));
		} else {
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		  int colIndex2 = (localIndexB[col(*icursor)] - nLocalInterior) *
		    nComponents + j;
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		  colsLocal.push_back(colIndex2);
		  valuesLocal.push_back(value(*icursor));
		}
	      }		
	    }


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	  }  // for each nnz in row
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	  if (rowPrimal) {
	    TEST_EXIT_DBG(globalPrimalIndex.count(*cursor))
	      ("Should not happen!\n");
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	    int rowIndex = globalPrimalIndex[*cursor] * nComponents + i;
	    MatSetValues(mat_primal_primal, 1, &rowIndex, cols.size(),
			 &(cols[0]), &(values[0]), ADD_VALUES);
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	    if (colsOther.size())
	      MatSetValues(mat_primal_b, 1, &rowIndex, colsOther.size(),
			   &(colsOther[0]), &(valuesOther[0]), ADD_VALUES);
	  } else {
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	    TEST_EXIT_DBG(localIndexB.count(*cursor))
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	      ("Should not happen!\n");
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	    int localRowIndex = localIndexB[*cursor] * nComponents + i; 
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	    for (unsigned int k = 0; k < cols.size(); k++)
	      cols[k] -= rStartB * nComponents;
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	    MatSetValues(mat_b_b, 1, &localRowIndex, cols.size(),
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			 &(cols[0]), &(values[0]), ADD_VALUES);
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	    if (colsOther.size()) {
	      int globalRowIndex = 
		(localIndexB[*cursor] + rStartB) * nComponents + i; 
	      MatSetValues(mat_b_primal, 1, &globalRowIndex, colsOther.size(),
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			   &(colsOther[0]), &(valuesOther[0]), ADD_VALUES);
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	    }
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	  }
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	  // === For preconditioner ===

	  if (!rowPrimal) {
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	    switch (fetiPreconditioner) {
	    case FETI_DIRICHLET:
	      {
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		int rowIndex = localIndexB[*cursor];
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		if (rowIndex < nLocalInterior) {
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		  int rowIndex2 = localIndexB[*cursor] * nComponents + i;
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		  MatSetValues(mat_interior_interior, 1, &rowIndex2, colsLocal.size(),
			       &(colsLocal[0]), &(valuesLocal[0]), INSERT_VALUES);
		  
		  if (colsLocalOther.size()) 
		    MatSetValues(mat_interior_duals, 1, &rowIndex2, colsLocalOther.size(),
				 &(colsLocalOther[0]), &(valuesLocalOther[0]), INSERT_VALUES);
		} else {
		  int rowIndex2 = 
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		    (localIndexB[*cursor] - nLocalInterior) * nComponents + i;
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		  MatSetValues(mat_duals_duals, 1, &rowIndex2, colsLocal.size(),
			       &(colsLocal[0]), &(valuesLocal[0]), INSERT_VALUES);
		  
		  if (colsLocalOther.size()) 
		    MatSetValues(mat_duals_interior, 1, &rowIndex2, colsLocalOther.size(),
				 &(colsLocalOther[0]), &(valuesLocalOther[0]), INSERT_VALUES);
		  
		}
	      }
	      break;
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	    case FETI_LUMPED:
	      {
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		int rowIndex = localIndexB[*cursor];
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		if (rowIndex >= nLocalInterior) {
		  int rowIndex2 = 
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		    (localIndexB[*cursor] - nLocalInterior) * nComponents + i;
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		  MatSetValues(mat_duals_duals, 1,