PreconditionerBlock.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: PreconditionerBlock.cpp
//
// For more information, please see: http://www.nektar.info
//
// The MIT License
//
// Copyright (c) 2006 Division of Applied Mathematics, Brown University (USA),
// Department of Aeronautics, Imperial College London (UK), and Scientific
// Computing and Imaging Institute, University of Utah (USA).
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
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// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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// DEALINGS IN THE SOFTWARE.
//
// Description: Block Preconditioner definition
//
///////////////////////////////////////////////////////////////////////////////
 
#include <LibUtilities/BasicUtils/VDmathArray.hpp>
#include <LocalRegions/MatrixKey.h>
#include <LocalRegions/SegExp.h>
#include <MultiRegions/AssemblyMap/AssemblyMapDG.h>
#include <MultiRegions/GlobalLinSys.h>
#include <MultiRegions/GlobalLinSysIterativeStaticCond.h>
#include <MultiRegions/GlobalMatrixKey.h>
#include <MultiRegions/PreconditionerBlock.h>
#include <cmath>
 
using namespace std;
 
namespace Nektar
{
namespace MultiRegions
{
/**
 * Registers the class with the Factory.
 */
string PreconditionerBlock::className =
    GetPreconFactory().RegisterCreatorFunction(
        "Block", PreconditionerBlock::create, "Block Diagonal Preconditioning");
/**
 * @class Block Preconditioner
 *
 * This class implements Block preconditioning for the conjugate
 * gradient matrix solver.
 */
 
PreconditionerBlock::PreconditionerBlock(
    const std::shared_ptr<GlobalLinSys> &plinsys,
    const AssemblyMapSharedPtr &pLocToGloMap)
    : Preconditioner(plinsys, pLocToGloMap)
{
}
 
void PreconditionerBlock::v_InitObject()
{
    GlobalSysSolnType solverType = m_locToGloMap.lock()->GetGlobalSysSolnType();
    m_isFull =
        (solverType == eIterativeFull) || (solverType == ePETScFullMatrix);
 
    ASSERTL0(solverType == MultiRegions::eIterativeStaticCond ||
                 solverType == MultiRegions::ePETScStaticCond || m_isFull,
             "Solver type not valid");
}
 
void PreconditionerBlock::v_BuildPreconditioner()
{
    GlobalLinSysKey key = m_linsys.lock()->GetKey();
 
    // Different setup for HDG and CG.
    if (key.GetMatrixType() == StdRegions::eHybridDGHelmBndLam)
    {
        BlockPreconditionerHDG();
    }
    else
    {
        BlockPreconditionerCG();
    }
}
 
/**
 * \brief Construct a block preconditioner from \f$\mathbf{S}_{1}\f$ for
 * the continuous Galerkin system.
 *
 * The preconditioner for the statically-condensed system is defined as:
 *
 * \f[\mathbf{M}^{-1}=\left[\begin{array}{ccc}
 *  \mathrm{Diag}[(\mathbf{S_{1}})_{vv}] & & \\
 *  & (\mathbf{S}_{1})_{eb} & \\
 *  & & (\mathbf{S}_{1})_{fb} \end{array}\right] \f]
 *
 * where \f$\mathbf{S}_{1}\f$ is the local Schur complement matrix for
 * each element and the subscript denotes the portion of the Schur
 * complement associated with the vertex (vv), edge (eb) and face
 * blocks (fb) respectively.
 *
 * In case of the full system \f$\mathbf{A}\f$, the preconditioner
 * includes the interior blocks as well. The full block preconditioner
 * is defined as:
 *
 * \f[\mathbf{M}^{-1}=\left[\begin{array}{cccc}
 *  \mathrm{Diag}[(\mathbf{A})_{vv}] & & & \\
 *  & (\mathbf{A})_{eb} & & \\
 *  & & (\mathbf{A})_{fb} & \\
 *  & & & (\mathbf{A})_{int} \end{array}\right] \f]
 *
 *  where the interior portion is added at the end following the
 *  ordering inside data structures.
 */
void PreconditionerBlock::BlockPreconditionerCG()
{
    ExpListSharedPtr expList = m_linsys.lock()->GetLocMat().lock();
    LocalRegions::ExpansionSharedPtr exp;
    DNekScalBlkMatSharedPtr loc_mat;
    DNekScalMatSharedPtr bnd_mat;
 
    int i, j, k, n, cnt, gId;
    int meshVertId, meshEdgeId, meshFaceId;
 
    auto asmMap = m_locToGloMap.lock();
 
    const int nExp    = expList->GetExpSize();
    const int nDirBnd = asmMap->GetNumGlobalDirBndCoeffs();
 
    // Grab periodic geometry information.
    PeriodicMap periodicVerts, periodicEdges, periodicFaces;
    expList->GetPeriodicEntities(periodicVerts, periodicEdges, periodicFaces);
 
    // The vectors below are of size 3(+1) to have separate storage for
    // vertices, edges and faces.
    // For the full matrix approach additional storage for the
    // interior matrices is required
    int nStorage = m_isFull ? 4 : 3;
 
    // Maps from geometry ID to the matrix representing the extracted
    // portion of S_1. For example idMats[2] folds the S_1 face blocks.
    vector<map<int, vector<NekDouble>>> idMats(nStorage);
 
    // Maps from the global ID, as obtained from AssemblyMapCG's
    // localToGlobalMap, to the geometry ID.
    vector<map<int, int>> gidMeshIds(3);
 
    // Maps from the global ID to the number of degrees of freedom for
    // this geometry object.
    vector<map<int, int>> gidDofs(nStorage);
 
    // Array containing maximum information needed for the universal
    // numbering later. For i = 0,1,2 for each geometry dimension:
    //
    // maxVertIds[2*i]   = maximum geometry ID at dimension i
    // maxVertIds[2*i+1] = maximum number of degrees of freedom for all
    //                     elements of dimension i.
    Array<OneD, int> maxVertIds(6, -1);
 
    // Figure out mapping from each elemental contribution to offset in
    // (vert,edge,face) triples.
    for (cnt = n = 0; n < nExp; ++n)
    {
        exp = expList->GetExp(n);
 
        // Grab reference to
        // StaticCond: Local Schur complement matrix
        // Full:       Boundary and interior matrix
        DNekScalMatSharedPtr schurMat;
 
        if (m_isFull)
        {
            // Get local matrix
            auto tmpMat = m_linsys.lock()->GetBlock(n);
 
            // Get number of boundary dofs
            int nBndDofs = exp->NumBndryCoeffs();
            int nIntDofs = exp->GetNcoeffs() - nBndDofs;
 
            // Get boundary & interior maps
            Array<OneD, unsigned int> bndMap(nBndDofs), intMap(nIntDofs);
            exp->GetBoundaryMap(bndMap);
            exp->GetInteriorMap(intMap);
 
            // Create temporary matrices
            DNekMatSharedPtr bndryMat =
                MemoryManager<DNekMat>::AllocateSharedPtr(nBndDofs, nBndDofs);
 
            // Extract boundary and send to StaticCond (Schur)
            // framework
            for (int i = 0; i < nBndDofs; ++i)
            {
                for (int j = 0; j < nBndDofs; ++j)
                {
                    (*bndryMat)(i, j) = (*tmpMat)(bndMap[i], bndMap[j]);
                }
            }
 
            schurMat =
                MemoryManager<DNekScalMat>::AllocateSharedPtr(1.0, bndryMat);
 
            // Extract interior matrix and save for block matrix setting
            vector<NekDouble> tmpStore(nIntDofs * nIntDofs);
            for (int i = 0; i < nIntDofs; ++i)
            {
                for (int j = 0; j < nIntDofs; ++j)
                {
                    tmpStore[j + i * nIntDofs] =
                        (*tmpMat)(intMap[i], intMap[j]);
                }
            }
 
            // Save interior mats and nDofs
            idMats[3][n]  = tmpStore;
            gidDofs[3][n] = nIntDofs;
            // then intMat gets added to a block diagonal matrix that
            // handles the extra interior dofs no need to do any
            // communication for that because all interior dofs are
            // local to each element
        }
        else
        {
            schurMat = m_linsys.lock()->GetStaticCondBlock(n)->GetBlock(0, 0);
        }
 
        // Process vertices to extract relevant portion of the Schur
        // complement matrix.
        for (i = 0; i < exp->GetNverts(); ++i)
        {
            meshVertId = exp->GetGeom()->GetVid(i);
            int locId  = exp->GetVertexMap(i);
 
            // Get the global ID of this vertex.
            gId = asmMap->GetLocalToGlobalMap(cnt + locId) - nDirBnd;
 
            // Ignore all Dirichlet vertices.
            if (gId < 0)
            {
                continue;
            }
 
            gidDofs[0][gId] = 1;
 
            // Extract vertex value from Schur complement matrix.
            NekDouble vertVal = (*schurMat)(locId, locId);
 
            // See if we have processed this vertex from another
            // element.
            auto gIt = idMats[0].find(gId);
 
            if (gIt == idMats[0].end())
            {
                // If not then put our 'matrix' inside idMats.
                idMats[0][gId] = vector<NekDouble>(1, vertVal);
            }
            else
            {
                // Otherwise combine with the value that is already
                // there (i.e. do assembly on this degree of freedom).
                gIt->second[0] += vertVal;
            }
 
            // Now check to see if the vertex is periodic. If it is,
            // then we change meshVertId to be the minimum of all the
            // other periodic vertices, so that we don't end up
            // duplicating the matrix in our final block matrix.
            auto pIt = periodicVerts.find(meshVertId);
            if (pIt != periodicVerts.end())
            {
                for (j = 0; j < pIt->second.size(); ++j)
                {
                    meshVertId = min(meshVertId, pIt->second[j].id);
                }
            }
 
            // Finally record the other information we need into the
            // other matrices.
            gidMeshIds[0][gId] = meshVertId;
            maxVertIds[0]      = max(maxVertIds[0], meshVertId);
            maxVertIds[1]      = 1;
        }
 
        // Process edges. This logic is mostly the same as the previous
        // block.
        for (i = 0; i < exp->GetGeom()->GetNumEdges(); ++i)
        {
            meshEdgeId = exp->GetGeom()->GetEid(i);
 
            Array<OneD, unsigned int> bmap, bmap2;
            Array<OneD, int> sign;
            StdRegions::Orientation edgeOrient = exp->GetGeom()->GetEorient(i);
 
            // Check if this edge is periodic. We may need to flip
            // orientation if it is.
            auto pIt = periodicEdges.find(meshEdgeId);
            if (pIt != periodicEdges.end())
            {
                pair<int, StdRegions::Orientation> idOrient =
                    DeterminePeriodicEdgeOrientId(meshEdgeId, edgeOrient,
                                                  pIt->second);
                meshEdgeId = idOrient.first;
                edgeOrient = idOrient.second;
            }
 
            // Grab edge interior map, and the edge inverse boundary
            // map, so that we can extract this edge from the Schur
            // complement matrix.
            if (exp->GetGeom()->GetNumFaces()) // 3D Element calls
            {
                exp->as<LocalRegions::Expansion3D>()
                    ->GetEdgeInteriorToElementMap(i, bmap, sign, edgeOrient);
                bmap2 = exp->as<LocalRegions::Expansion3D>()
                            ->GetEdgeInverseBoundaryMap(i);
            }
            else
            {
                exp->GetTraceInteriorToElementMap(i, bmap, sign, edgeOrient);
                bmap2 = exp->as<LocalRegions::Expansion2D>()
                            ->GetTraceInverseBoundaryMap(i);
            }
 
            // Allocate temporary storage for the extracted edge matrix.
            const int nEdgeCoeffs = bmap.size();
            vector<NekDouble> tmpStore(nEdgeCoeffs * nEdgeCoeffs);
 
            gId = asmMap->GetLocalToGlobalMap(cnt + bmap[0]);
 
            for (j = 0; j < nEdgeCoeffs; ++j)
            {
                // We record the minimum ID from the edge for our
                // maps. This follows the logic that the assembly map
                // ordering will always give us a contiguous ordering of
                // global degrees of freedom for edge interior
                // coefficients.
                gId = min(gId,
                          asmMap->GetLocalToGlobalMap(cnt + bmap[j]) - nDirBnd);
 
                // Ignore Dirichlet edges.
                if (gId < 0)
                {
                    continue;
                }
 
                const NekDouble sign1 = sign[j];
 
                // Extract this edge, along with sign array for assembly
                // later.
                for (k = 0; k < nEdgeCoeffs; ++k)
                {
                    tmpStore[k + j * nEdgeCoeffs] =
                        sign1 * sign[k] * (*schurMat)(bmap2[j], bmap2[k]);
                }
            }
 
            if (gId < 0)
            {
                continue;
            }
 
            gidDofs[1][gId] = nEdgeCoeffs;
 
            // Assemble this edge matrix with another one, if it exists.
            auto gIt = idMats[1].find(gId);
            if (gIt == idMats[1].end())
            {
                idMats[1][gId] = tmpStore;
            }
            else
            {
                ASSERTL1(tmpStore.size() == gIt->second.size(),
                         "Number of modes mismatch");
                Vmath::Vadd(nEdgeCoeffs * nEdgeCoeffs, &gIt->second[0], 1,
                            &tmpStore[0], 1, &gIt->second[0], 1);
            }
 
            gidMeshIds[1][gId] = meshEdgeId;
            maxVertIds[2]      = max(maxVertIds[2], meshEdgeId);
            maxVertIds[3]      = max(maxVertIds[3], nEdgeCoeffs);
        }
 
        // Process faces. This logic is mostly the same as the previous
        // block.
        for (i = 0; i < exp->GetGeom()->GetNumFaces(); ++i)
        {
            meshFaceId = exp->GetGeom()->GetFid(i);
 
            Array<OneD, unsigned int> bmap, bmap2;
            Array<OneD, int> sign;
            StdRegions::Orientation faceOrient = exp->GetGeom()->GetForient(i);
 
            // Check if this face is periodic. We may need to flip
            // orientation if it is.
            auto pIt = periodicFaces.find(meshFaceId);
            if (pIt != periodicFaces.end())
            {
                meshFaceId = min(meshFaceId, pIt->second[0].id);
                faceOrient = DeterminePeriodicFaceOrient(faceOrient,
                                                         pIt->second[0].orient);
            }
 
            exp->GetTraceInteriorToElementMap(i, bmap, sign, faceOrient);
            bmap2 = exp->as<LocalRegions::Expansion3D>()
                        ->GetTraceInverseBoundaryMap(i);
 
            // Allocate temporary storage for the extracted face matrix.
            const int nFaceCoeffs = bmap.size();
            vector<NekDouble> tmpStore(nFaceCoeffs * nFaceCoeffs);
 
            gId = asmMap->GetLocalToGlobalMap(cnt + bmap[0]);
 
            for (j = 0; j < nFaceCoeffs; ++j)
            {
                gId = min(gId,
                          asmMap->GetLocalToGlobalMap(cnt + bmap[j]) - nDirBnd);
 
                // Ignore Dirichlet faces.
                if (gId < 0)
                {
                    continue;
                }
 
                const NekDouble sign1 = sign[j];
 
                // Extract this face, along with sign array for assembly
                // later.
                for (k = 0; k < nFaceCoeffs; ++k)
                {
                    tmpStore[k + j * nFaceCoeffs] =
                        sign1 * sign[k] * (*schurMat)(bmap2[j], bmap2[k]);
                }
            }
 
            if (gId < 0)
            {
                continue;
            }
 
            gidDofs[2][gId] = nFaceCoeffs;
 
            // Assemble this face matrix with another one, if it exists.
            auto gIt = idMats[2].find(gId);
            if (gIt == idMats[2].end())
            {
                idMats[2][gId] = tmpStore;
            }
            else
            {
                ASSERTL1(tmpStore.size() == gIt->second.size(),
                         "Number of modes mismatch");
                Vmath::Vadd(nFaceCoeffs * nFaceCoeffs, &gIt->second[0], 1,
                            &tmpStore[0], 1, &gIt->second[0], 1);
            }
 
            gidMeshIds[2][gId] = meshFaceId;
            maxVertIds[4]      = max(maxVertIds[4], meshFaceId);
            maxVertIds[5]      = max(maxVertIds[5], nFaceCoeffs);
        }
 
        cnt += exp->GetNcoeffs();
    }
 
    // Perform a reduction to find maximum vertex, edge and face
    // geometry IDs.
    m_comm = expList->GetSession()->GetComm()->GetRowComm();
    m_comm->AllReduce(maxVertIds, LibUtilities::ReduceMax);
 
    // Concatenate all matrices into contiguous storage and figure out
    // universal ID numbering.
    vector<NekDouble> storageBuf;
    vector<long> globalToUniversal;
 
    for (i = 0, cnt = 1; i < 3; ++i)
    {
        const int maxDofs = maxVertIds[2 * i + 1];
 
        // Note that iterating over the map uses the property that keys
        // are accessed in order of ascending order, putting everything
        // in the right order for the global system.
        for (auto &gIt : idMats[i])
        {
            // Copy matrix into storage.
            storageBuf.insert(storageBuf.end(), gIt.second.begin(),
                              gIt.second.end());
 
            // Get mesh ID from global ID number.
            ASSERTL1(gidMeshIds[i].count(gIt.first) > 0,
                     "Unable to find global ID " +
                         boost::lexical_cast<string>(gIt.first) +
                         " inside map");
            meshVertId = gidMeshIds[i][gIt.first];
 
            for (j = 0; j < gIt.second.size(); ++j)
            {
                globalToUniversal.push_back(cnt +
                                            meshVertId * maxDofs * maxDofs + j);
            }
 
            // Free up the temporary storage.
            gIt.second.clear();
        }
 
        cnt += (maxVertIds[2 * i] + 1) * maxDofs * maxDofs;
    }
 
    ASSERTL1(storageBuf.size() == globalToUniversal.size(),
             "Storage buffer and global to universal map size does "
             "not match");
 
    Array<OneD, NekDouble> storageData(storageBuf.size(), &storageBuf[0]);
    Array<OneD, long> globalToUniversalMap(globalToUniversal.size(),
                                           &globalToUniversal[0]);
 
    // Use GS to assemble data between processors.
    Gs::gs_data *tmpGs =
        Gs::Init(globalToUniversalMap, m_comm,
                 expList->GetSession()->DefinesCmdLineArgument("verbose"));
    Gs::Gather(storageData, Gs::gs_add, tmpGs);
 
    // Figure out what storage we need in the block matrix.
    int nblksSize = 1 + idMats[1].size() + idMats[2].size();
    if (m_isFull)
    {
        nblksSize += idMats[3].size();
    }
    Array<OneD, unsigned int> n_blks(nblksSize);
 
    // Vertex block is a diagonal matrix.
    n_blks[0] = idMats[0].size();
 
    // Now extract number of rows in each edge and face block from the
    // gidDofs map.
    cnt = 1;
    for (i = 1; i < nStorage; ++i)
    {
        for (auto &gIt : idMats[i])
        {
            ASSERTL1(gidDofs[i].count(gIt.first) > 0,
                     "Unable to find number of degrees of freedom for "
                     "global ID " +
                         boost::lexical_cast<string>(gIt.first));
 
            n_blks[cnt++] = gidDofs[i][gIt.first];
        }
    }
 
    // Allocate storage for the block matrix.
    m_blkMat =
        MemoryManager<DNekBlkMat>::AllocateSharedPtr(n_blks, n_blks, eDIAGONAL);
 
    // We deal with the vertex matrix separately since all vertices can
    // be condensed into a single, block-diagonal matrix.
    DNekMatSharedPtr vertMat = MemoryManager<DNekMat>::AllocateSharedPtr(
        n_blks[0], n_blks[0], 0.0, eDIAGONAL);
 
    // Fill the vertex matrix with the inverse of each vertex value.
    cnt = 0;
    for (auto gIt = idMats[0].begin(); gIt != idMats[0].end(); ++gIt, ++cnt)
    {
        (*vertMat)(cnt, cnt) = 1.0 / storageData[cnt];
    }
 
    // Put the vertex matrix in the block matrix.
    m_blkMat->SetBlock(0, 0, vertMat);
 
    // Finally, grab the matrices from the block storage, invert them
    // and place them in the correct position inside the block matrix.
    int cnt2 = 1;
    for (i = 1; i < nStorage; ++i)
    {
        for (auto &gIt : idMats[i])
        {
            int nDofs = gidDofs[i][gIt.first];
 
            DNekMatSharedPtr tmp =
                MemoryManager<DNekMat>::AllocateSharedPtr(nDofs, nDofs);
 
            for (j = 0; j < nDofs; ++j)
            {
                for (k = 0; k < nDofs; ++k)
                {
                    // Interior matrices do not need mapping
                    // and are stored in idMats[3]
                    if (m_isFull && i == 3)
                    {
                        (*tmp)(j, k) = gIt.second[k + j * nDofs];
                    }
                    // Boundary matrices
                    else
                    {
                        (*tmp)(j, k) = storageData[k + j * nDofs + cnt];
                    }
                }
            }
 
            cnt += nDofs * nDofs;
 
            tmp->Invert();
            m_blkMat->SetBlock(cnt2, cnt2, tmp);
            ++cnt2;
        }
    }
}
 
/**
 * @brief Construct a block preconditioner for the hybridized
 * discontinuous Galerkin system.
 *
 * This system is built in a similar fashion to its continuous variant
 * found in PreconditionerBlock::BlockPreconditionerCG. In this setting
 * however, the matrix is constructed as:
 *
 * \f[ M^{-1} = \mathrm{Diag}[ (\mathbf{S_{1}})_{f}^{-1} ] \f]
 *
 * where each matrix is the Schur complement system restricted to a
 * single face of the trace system.
 */
void PreconditionerBlock::BlockPreconditionerHDG()
{
    std::shared_ptr<MultiRegions::ExpList> expList =
        ((m_linsys.lock())->GetLocMat()).lock();
    std::shared_ptr<MultiRegions::ExpList> trace = expList->GetTrace();
    LocalRegions::ExpansionSharedPtr locExpansion;
    DNekScalBlkMatSharedPtr loc_mat;
    DNekScalMatSharedPtr bnd_mat;
 
    AssemblyMapDGSharedPtr asmMap =
        std::dynamic_pointer_cast<AssemblyMapDG>(m_locToGloMap.lock());
 
    int i, j, k, n, cnt, cnt2;
 
    // Figure out number of Dirichlet trace elements
    int nTrace = expList->GetTrace()->GetExpSize();
    int nDir   = asmMap->GetNumGlobalDirBndCoeffs();
 
    for (cnt = n = 0; n < nTrace; ++n)
    {
        if (cnt >= nDir)
        {
            break;
        }
 
        cnt += trace->GetExp(n)->GetNcoeffs();
    }
 
    nDir = n;
 
    // Allocate storage for block matrix. Need number of unique faces in
    // trace space.
    int maxTraceSize = -1;
    map<int, int> arrayOffsets;
 
    for (cnt = 0, n = nDir; n < nTrace; ++n)
    {
        int nCoeffs  = trace->GetExp(n)->GetNcoeffs();
        int nCoeffs2 = nCoeffs * nCoeffs;
 
        arrayOffsets[n] = cnt;
        cnt += nCoeffs2;
 
        if (maxTraceSize < nCoeffs2)
        {
            maxTraceSize = nCoeffs2;
        }
    }
 
    // Find maximum trace size.
    m_comm = expList->GetSession()->GetComm()->GetRowComm();
    m_comm->AllReduce(maxTraceSize, LibUtilities::ReduceMax);
 
    // Zero matrix storage.
    Array<OneD, NekDouble> tmpStore(cnt, 0.0);
 
    // Assemble block matrices for each trace element.
    for (cnt = n = 0; n < expList->GetExpSize(); ++n)
    {
        int elmt     = n;
        locExpansion = expList->GetExp(elmt);
 
        Array<OneD, LocalRegions::ExpansionSharedPtr> &elmtToTraceMap =
            asmMap->GetElmtToTrace()[elmt];
 
        // Block matrix (lambda)
        loc_mat = (m_linsys.lock())->GetStaticCondBlock(n);
        bnd_mat = loc_mat->GetBlock(0, 0);
 
        int nFacets = locExpansion->GetNtraces();
 
        for (cnt2 = i = 0; i < nFacets; ++i)
        {
            int nCoeffs = elmtToTraceMap[i]->GetNcoeffs();
            int elmtId  = elmtToTraceMap[i]->GetElmtId();
 
            // Ignore Dirichlet edges.
            if (elmtId < nDir)
            {
                cnt += nCoeffs;
                cnt2 += nCoeffs;
                continue;
            }
 
            NekDouble *off = &tmpStore[arrayOffsets[elmtId]];
 
            for (j = 0; j < nCoeffs; ++j)
            {
                NekDouble sign1 = asmMap->GetLocalToGlobalBndSign(cnt + j);
                for (k = 0; k < nCoeffs; ++k)
                {
                    NekDouble sign2 = asmMap->GetLocalToGlobalBndSign(cnt + k);
                    off[j * nCoeffs + k] +=
                        (*bnd_mat)(cnt2 + j, cnt2 + k) * sign1 * sign2;
                }
            }
 
            cnt += nCoeffs;
            cnt2 += nCoeffs;
        }
    }
 
    // Set up IDs for universal numbering.
    Array<OneD, long> uniIds(tmpStore.size());
    for (cnt = 0, n = nDir; n < nTrace; ++n)
    {
        LocalRegions::ExpansionSharedPtr traceExp = trace->GetExp(n);
        int nCoeffs                               = traceExp->GetNcoeffs();
        int geomId = traceExp->GetGeom()->GetGlobalID();
 
        for (i = 0; i < nCoeffs * nCoeffs; ++i)
        {
            uniIds[cnt++] = geomId * maxTraceSize + i + 1;
        }
    }
 
    // Assemble matrices across partitions.
    Gs::gs_data *gsh =
        Gs::Init(uniIds, m_comm,
                 expList->GetSession()->DefinesCmdLineArgument("verbose"));
    Gs::Gather(tmpStore, Gs::gs_add, gsh);
 
    // Set up diagonal block matrix
    Array<OneD, unsigned int> n_blks(nTrace - nDir);
    for (n = 0; n < nTrace - nDir; ++n)
    {
        n_blks[n] = trace->GetExp(n + nDir)->GetNcoeffs();
    }
 
    m_blkMat =
        MemoryManager<DNekBlkMat>::AllocateSharedPtr(n_blks, n_blks, eDIAGONAL);
 
    for (n = 0; n < nTrace - nDir; ++n)
    {
        int nCoeffs = trace->GetExp(n + nDir)->GetNcoeffs();
        DNekMatSharedPtr tmp =
            MemoryManager<DNekMat>::AllocateSharedPtr(nCoeffs, nCoeffs);
        NekDouble *off = &tmpStore[arrayOffsets[n + nDir]];
 
        for (i = 0; i < nCoeffs; ++i)
        {
            for (j = 0; j < nCoeffs; ++j)
            {
                (*tmp)(i, j) = off[i * nCoeffs + j];
            }
        }
 
        tmp->Invert();
        m_blkMat->SetBlock(n, n, tmp);
    }
}
 
/**
 * @brief Apply preconditioner to \p pInput and store the result in \p
 * pOutput.
 */
void PreconditionerBlock::v_DoPreconditioner(
    const Array<OneD, NekDouble> &pInput, Array<OneD, NekDouble> &pOutput)
{
    // Get assembly map and solver type
    auto asmMap = m_locToGloMap.lock();
 
    // Determine matrix size
    int nGlobal = m_isFull ? asmMap->GetNumGlobalCoeffs()
                           : asmMap->GetNumGlobalBndCoeffs();
    int nDir    = asmMap->GetNumGlobalDirBndCoeffs();
    int nNonDir = nGlobal - nDir;
 
    // Apply preconditioner
    DNekBlkMat &M = (*m_blkMat);
    NekVector<NekDouble> r(nNonDir, pInput, eWrapper);
    NekVector<NekDouble> z(nNonDir, pOutput, eWrapper);
    z = M * r;
}
} // namespace MultiRegions
} // namespace Nektar