AssemblyMapCG.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: AssemblyMapCG.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
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included
// in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
// THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
// DEALINGS IN THE SOFTWARE.
//
// Description: C0-continuous Local to Global mapping routines, base class
//
///////////////////////////////////////////////////////////////////////////////
 
#include <LibUtilities/BasicUtils/HashUtils.hpp>
#include <LibUtilities/BasicUtils/ShapeType.hpp>
#include <LocalRegions/Expansion.h>
#include <LocalRegions/Expansion2D.h>
#include <LocalRegions/Expansion3D.h>
#include <MultiRegions/AssemblyMap/AssemblyMapCG.h>
#include <MultiRegions/ExpList.h>
 
#include <boost/config.hpp>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/bandwidth.hpp>
#include <boost/graph/cuthill_mckee_ordering.hpp>
#include <boost/graph/properties.hpp>
 
using namespace std;
 
namespace Nektar::MultiRegions
{
/**
 * @class AssemblyMapCG
 * Mappings are created for three possible global solution types:
 *  - Direct full matrix
 *  - Direct static condensation
 *  - Direct multi-level static condensation
 * In the latter case, mappings are created recursively for the
 * different levels of static condensation.
 *
 * These mappings are used by GlobalLinSys to generate the global
 * system.
 */
 
/**
 *
 */
AssemblyMapCG::AssemblyMapCG(
    const LibUtilities::SessionReaderSharedPtr &pSession,
    const LibUtilities::CommSharedPtr &comm, const std::string variable)
    : AssemblyMap(pSession, comm, variable)
{
    pSession->LoadParameter("MaxStaticCondLevel", m_maxStaticCondLevel, 100);
}
 
int AssemblyMapCG::CreateGraph(
    const ExpList &locExp, const BndCondExp &bndCondExp,
    const Array<OneD, const BndCond> &bndConditions,
    const bool checkIfSystemSingular, const PeriodicMap &periodicVerts,
    const PeriodicMap &periodicEdges, const PeriodicMap &periodicFaces,
    DofGraph &graph, BottomUpSubStructuredGraphSharedPtr &bottomUpGraph,
    set<int> &extraDirVerts, set<int> &extraDirEdges,
    int &firstNonDirGraphVertId, int &nExtraDirichlet, int mdswitch)
{
    int graphVertId = 0;
    int vMaxVertId  = -1;
    int i, j, k, l, cnt;
    int meshVertId, meshEdgeId, meshFaceId;
    int meshVertId2, meshEdgeId2;
 
    LocalRegions::ExpansionSharedPtr exp, bndExp;
    const LocalRegions::ExpansionVector &locExpVector = *(locExp.GetExp());
    LibUtilities::CommSharedPtr vComm                 = m_comm->GetRowComm();
 
    m_numLocalBndCondCoeffs = 0;
    m_systemSingular        = checkIfSystemSingular;
 
    for (i = 0; i < bndCondExp.size(); i++)
    {
 
        m_numLocalBndCondCoeffs += bndCondExp[i]->GetNcoeffs();
 
        if (bndConditions[0][i]->GetBoundaryConditionType() ==
            SpatialDomains::ePeriodic)
        {
            continue;
        }
 
        // Check to see if any value on boundary has Dirichlet
        // value.  note this is a vector to manage coupled
        // solver but for scalar will just be a vector of size 11
        cnt = 0;
        for (k = 0; k < bndConditions.size(); ++k)
        {
            if (bndConditions[k][i]->GetBoundaryConditionType() ==
                SpatialDomains::eDirichlet)
            {
                cnt++;
            }
            if (bndConditions[k][i]->GetBoundaryConditionType() !=
                SpatialDomains::eNeumann)
            {
                m_systemSingular = false;
            }
        }
 
        // Find the maximum boundary vertex ID on this process. This is
        // used later to pin a vertex if the system is singular.
        for (j = 0; j < bndCondExp[i]->GetNumElmts(); ++j)
        {
            bndExp = bndCondExp[i]->GetExp(j)->as<LocalRegions::Expansion>();
            for (k = 0; k < bndExp->GetNverts(); ++k)
            {
                if (vMaxVertId < bndExp->GetGeom()->GetVid(k))
                {
                    vMaxVertId = bndExp->GetGeom()->GetVid(k);
                }
            }
        }
 
        // If all boundaries are Dirichlet fill in graph
        if (cnt == bndConditions.size())
        {
            for (j = 0; j < bndCondExp[i]->GetNumElmts(); j++)
            {
                bndExp = bndCondExp[i]->GetExp(j);
 
                for (k = 0; k < bndExp->GetNverts(); k++)
                {
                    meshVertId = bndExp->GetGeom()->GetVid(k);
                    if (graph[0].count(meshVertId) == 0)
                    {
                        graph[0][meshVertId] = graphVertId++;
                    }
                }
 
                const int bndDim = bndExp->GetNumBases();
                if (bndDim > 1)
                {
                    for (k = 0; k < bndExp->GetNtraces(); k++)
                    {
                        meshEdgeId = bndExp->GetGeom()->GetEid(k);
                        if (graph[1].count(meshEdgeId) == 0)
                        {
                            graph[1][meshEdgeId] = graphVertId++;
                        }
                    }
                }
 
                // Possibility of a face in 3D or edge in 2D
                meshFaceId = bndExp->GetGeom()->GetGlobalID();
                if (graph[bndDim].count(meshFaceId) == 0)
                {
                    graph[bndDim][meshFaceId] = graphVertId++;
                }
                m_numLocalDirBndCoeffs += bndExp->GetNcoeffs();
            }
        }
    }
 
    // Number of dirichlet edges and faces (not considering periodic
    // BCs)
    m_numDirEdges = graph[1].size();
    m_numDirFaces = graph[2].size();
 
    /*
     * The purpose of this routine is to deal with those degrees of
     * freedom that are Dirichlet, but do not have a local Dirichlet
     * boundary condition expansion set.
     *
     * For example, in 2D, consider a triangulation of a square into two
     * triangles. Now imagine one edge of the square is Dirichlet and
     * the problem is run on two processors. On one processor, one
     * triangle vertex is Dirichlet, but doesn't know this since the
     * Dirichlet composite lives on the other processor.
     *
     * When the global linear system is solved therefore, there is an
     * inconsistency that at best leads to an inaccurate answer or a
     * divergence of the system.
     *
     * This routine identifies such cases for 2D, and also for 3D where
     * e.g. edges may have the same problem (consider an extrusion of
     * the case above, for example).
     */
 
    // Collate information on Dirichlet vertices from all processes
    int n = vComm->GetSize();
    int p = vComm->GetRank();
 
    if (vComm->IsSerial())
    {
        // for FieldConvert Comm this is true and it resets
        // parallel processing back to serial case
        n = 1;
        p = 0;
    }
    // At this point, graph only contains information from Dirichlet
    // boundaries. Therefore make a global list of the vert and edge
    // information on all processors.
    Array<OneD, int> vertcounts(n, 0);
    Array<OneD, int> vertoffsets(n, 0);
    Array<OneD, int> edgecounts(n, 0);
    Array<OneD, int> edgeoffsets(n, 0);
    vertcounts[p] = graph[0].size();
    edgecounts[p] = graph[1].size();
    vComm->AllReduce(vertcounts, LibUtilities::ReduceSum);
    vComm->AllReduce(edgecounts, LibUtilities::ReduceSum);
 
    for (i = 1; i < n; ++i)
    {
        vertoffsets[i] = vertoffsets[i - 1] + vertcounts[i - 1];
        edgeoffsets[i] = edgeoffsets[i - 1] + edgecounts[i - 1];
    }
 
    int nTotVerts = Vmath::Vsum(n, vertcounts, 1);
    int nTotEdges = Vmath::Vsum(n, edgecounts, 1);
 
    Array<OneD, int> vertlist(nTotVerts, 0);
    Array<OneD, int> edgelist(nTotEdges, 0);
 
    // construct list of global ids of global vertices
    i = 0;
    for (auto &it : graph[0])
    {
        vertlist[vertoffsets[p] + i++] = it.first;
    }
 
    // construct list of global ids of global edges
    i = 0;
    for (auto &it : graph[1])
    {
        edgelist[edgeoffsets[p] + i++] = it.first;
    }
    vComm->AllReduce(vertlist, LibUtilities::ReduceSum);
    vComm->AllReduce(edgelist, LibUtilities::ReduceSum);
 
    // Now we have a list of all Dirichlet vertices and edges on all
    // processors.
    nExtraDirichlet = 0;
    map<int, int> extraDirVertIds, extraDirEdgeIds;
 
    // Ensure Dirchlet vertices are consistently recorded between
    // processes (e.g. Dirichlet region meets Neumann region across a
    // partition boundary requires vertex on partition to be Dirichlet).
    //
    // To do this we look over all elements and vertices in local
    // partition and see if they match the values stored in the vertlist
    // from other processors and if so record the meshVertId/meshEdgeId
    // and the processor it comes from.
    for (i = 0; i < n; ++i)
    {
        if (i == p)
        {
            continue;
        }
 
        for (j = 0; j < locExpVector.size(); j++)
        {
            exp = locExpVector[j];
 
            for (k = 0; k < exp->GetNverts(); k++)
            {
                meshVertId = exp->GetGeom()->GetVid(k);
                if (graph[0].count(meshVertId) == 0)
                {
                    for (l = 0; l < vertcounts[i]; ++l)
                    {
                        if (vertlist[vertoffsets[i] + l] == meshVertId)
                        {
                            extraDirVertIds[meshVertId] = i;
                            graph[0][meshVertId]        = graphVertId++;
                            nExtraDirichlet++;
                        }
                    }
                }
            }
 
            for (k = 0; k < exp->GetGeom()->GetNumEdges(); k++)
            {
                meshEdgeId = exp->GetGeom()->GetEid(k);
                if (graph[1].count(meshEdgeId) == 0)
                {
                    for (l = 0; l < edgecounts[i]; ++l)
                    {
                        if (edgelist[edgeoffsets[i] + l] == meshEdgeId)
                        {
                            extraDirEdgeIds[meshEdgeId] = i;
                            graph[1][meshEdgeId]        = graphVertId++;
                            if (exp->GetGeom()->GetNumFaces())
                            {
                                nExtraDirichlet +=
                                    exp->as<LocalRegions::Expansion3D>()
                                        ->GetEdgeNcoeffs(k) -
                                    2;
                            }
                            else
                            {
                                nExtraDirichlet += exp->GetTraceNcoeffs(k) - 2;
                            }
                        }
                    }
                }
            }
        }
    }
 
    // Low Energy preconditioner needs to know how many extra Dirichlet
    // edges are on this process so store map in array.
    m_extraDirEdges = Array<OneD, int>(extraDirEdgeIds.size(), -1);
    i               = 0;
    for (auto &it : extraDirEdgeIds)
    {
        meshEdgeId           = it.first;
        m_extraDirEdges[i++] = meshEdgeId;
    }
 
    // Now we have a list of all vertices and edges that are Dirichlet
    // and not defined on the local partition as well as which processor
    // they are stored on.
    //
    // Make a full list of all such entities on all processors and which
    // processor they belong to.
    for (i = 0; i < n; ++i)
    {
        vertcounts[i]  = 0;
        vertoffsets[i] = 0;
        edgecounts[i]  = 0;
        edgeoffsets[i] = 0;
    }
 
    vertcounts[p] = extraDirVertIds.size();
    edgecounts[p] = extraDirEdgeIds.size();
    vComm->AllReduce(vertcounts, LibUtilities::ReduceSum);
    vComm->AllReduce(edgecounts, LibUtilities::ReduceSum);
    nTotVerts = Vmath::Vsum(n, vertcounts, 1);
    nTotEdges = Vmath::Vsum(n, edgecounts, 1);
 
    vertoffsets[0] = edgeoffsets[0] = 0;
 
    for (i = 1; i < n; ++i)
    {
        vertoffsets[i] = vertoffsets[i - 1] + vertcounts[i - 1];
        edgeoffsets[i] = edgeoffsets[i - 1] + edgecounts[i - 1];
    }
 
    Array<OneD, int> vertids(nTotVerts, 0);
    Array<OneD, int> edgeids(nTotEdges, 0);
    Array<OneD, int> vertprocs(nTotVerts, 0);
    Array<OneD, int> edgeprocs(nTotEdges, 0);
 
    i = 0;
    for (auto &it : extraDirVertIds)
    {
        vertids[vertoffsets[p] + i]   = it.first;
        vertprocs[vertoffsets[p] + i] = it.second;
        ++i;
    }
 
    i = 0;
    for (auto &it : extraDirEdgeIds)
    {
        edgeids[edgeoffsets[p] + i]   = it.first;
        edgeprocs[edgeoffsets[p] + i] = it.second;
        ++i;
    }
 
    vComm->AllReduce(vertids, LibUtilities::ReduceSum);
    vComm->AllReduce(vertprocs, LibUtilities::ReduceSum);
    vComm->AllReduce(edgeids, LibUtilities::ReduceSum);
    vComm->AllReduce(edgeprocs, LibUtilities::ReduceSum);
 
    // Set up list of vertices that need to be shared to other
    // partitions
    for (i = 0; i < nTotVerts; ++i)
    {
        if (p == vertprocs[i]) // rank = vertproc[i]
        {
            extraDirVerts.insert(vertids[i]);
        }
    }
 
    // Set up list of edges that need to be shared to other partitions
    for (i = 0; i < nTotEdges; ++i)
    {
        if (p == edgeprocs[i]) // rank = vertproc[i]
        {
            extraDirEdges.insert(edgeids[i]);
        }
    }
 
    // Check between processes if the whole system is singular
    int s = m_systemSingular ? 1 : 0;
    vComm->AllReduce(s, LibUtilities::ReduceMin);
    m_systemSingular = s == 1 ? true : false;
 
    // Find the minimum boundary vertex ID on each process
    Array<OneD, int> bcminvertid(n, 0);
    bcminvertid[p] = vMaxVertId;
    vComm->AllReduce(bcminvertid, LibUtilities::ReduceMax);
 
    // Find the process rank with the minimum boundary vertex ID
    int maxIdx = Vmath::Imax(n, bcminvertid, 1);
 
    // If the system is singular, the process with the maximum
    // number of BCs will set a Dirichlet vertex to make
    // system non-singular.  Note: we find the process with
    // maximum boundary regions to ensure we do not try to set
    // a Dirichlet vertex on a partition with no intersection
    // with the boundary.
    meshVertId = 0;
 
    if (m_systemSingular && checkIfSystemSingular && maxIdx == p)
    {
        if (m_session->DefinesParameter("SingularVertex"))
        {
            m_session->LoadParameter("SingularVertex", meshVertId);
        }
        else if (vMaxVertId == -1)
        {
            // All boundaries are periodic.
            meshVertId = locExpVector[0]->GetGeom()->GetVid(0);
        }
        else
        {
            // Set pinned vertex to that with minimum vertex ID to
            // ensure consistency in parallel.
            meshVertId = bcminvertid[p];
        }
 
        if (graph[0].count(meshVertId) == 0)
        {
            graph[0][meshVertId] = graphVertId++;
        }
    }
 
    vComm->AllReduce(meshVertId, LibUtilities::ReduceSum);
 
    // When running in parallel, we need to ensure that the singular
    // mesh vertex is communicated to any periodic vertices, otherwise
    // the system may diverge.
    if (m_systemSingular && checkIfSystemSingular)
    {
        // Firstly, we check that no other processors have this
        // vertex. If they do, then we mark the vertex as also being
        // Dirichlet.
        if (maxIdx != p)
        {
            for (i = 0; i < locExpVector.size(); ++i)
            {
                for (j = 0; j < locExpVector[i]->GetNverts(); ++j)
                {
                    if (locExpVector[i]->GetGeom()->GetVid(j) != meshVertId)
                    {
                        continue;
                    }
 
                    if (graph[0].count(meshVertId) == 0)
                    {
                        graph[0][meshVertId] = graphVertId++;
                    }
                }
            }
        }
 
        // In the case that meshVertId is periodic with other vertices,
        // this process and all other processes need to make sure that
        // the periodic vertices are also marked as Dirichlet.
        int gId;
 
        // At least one process (maxBCidx) will have already associated
        // a graphVertId with meshVertId. Others won't even have any of
        // the vertices. The logic below is designed to handle both
        // cases.
        if (graph[0].count(meshVertId) == 0)
        {
            gId = -1;
        }
        else
        {
            gId = graph[0][meshVertId];
        }
 
        for (auto &pIt : periodicVerts)
        {
            // Either the vertex is local to this processor (in which
            // case it will be in the pIt.first position) or else
            // meshVertId might be contained within another processor's
            // vertex list. The if statement below covers both cases. If
            // we find it, set as Dirichlet with the vertex id gId.
            if (pIt.first == meshVertId)
            {
                gId                  = gId < 0 ? graphVertId++ : gId;
                graph[0][meshVertId] = gId;
 
                for (i = 0; i < pIt.second.size(); ++i)
                {
                    if (pIt.second[i].isLocal)
                    {
                        graph[0][pIt.second[i].id] = graph[0][meshVertId];
                    }
                }
            }
            else
            {
                bool found = false;
                for (i = 0; i < pIt.second.size(); ++i)
                {
                    if (pIt.second[i].id == meshVertId)
                    {
                        found = true;
                        break;
                    }
                }
 
                if (found)
                {
                    gId                 = gId < 0 ? graphVertId++ : gId;
                    graph[0][pIt.first] = gId;
 
                    for (i = 0; i < pIt.second.size(); ++i)
                    {
                        if (pIt.second[i].isLocal)
                        {
                            graph[0][pIt.second[i].id] = graph[0][pIt.first];
                        }
                    }
                }
            }
        }
    }
 
    // Add extra dirichlet boundary conditions to count.
    m_numLocalDirBndCoeffs += nExtraDirichlet;
    firstNonDirGraphVertId = graphVertId;
 
    typedef boost::adjacency_list<boost::setS, boost::vecS, boost::undirectedS>
        BoostGraph;
    BoostGraph boostGraphObj;
 
    vector<map<int, int>> tempGraph(3);
    map<int, int> vwgts_map;
    Array<OneD, int> localVerts;
    Array<OneD, int> localEdges;
    Array<OneD, int> localFaces;
 
    int tempGraphVertId = 0;
    int localVertOffset = 0;
    int localEdgeOffset = 0;
    int localFaceOffset = 0;
    int nTotalVerts     = 0;
    int nTotalEdges     = 0;
    int nTotalFaces     = 0;
    int nVerts;
    int nEdges;
    int nFaces;
    int vertCnt;
    int edgeCnt;
    int faceCnt;
 
    m_numNonDirVertexModes = 0;
    m_numNonDirEdges       = 0;
    m_numNonDirFaces       = 0;
    m_numNonDirFaceModes   = 0;
    m_numNonDirFaceModes   = 0;
    m_numLocalBndCoeffs    = 0;
 
    map<int, int> EdgeSize;
    map<int, int> FaceSize;
 
    /// -  Count verts, edges, face and add up edges and face sizes
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp    = locExpVector[i];
        nEdges = exp->GetGeom()->GetNumEdges();
        nFaces = exp->GetGeom()->GetNumFaces();
 
        nTotalVerts += exp->GetNverts();
        nTotalEdges += nEdges;
        nTotalFaces += nFaces;
 
        for (j = 0; j < nEdges; ++j)
        {
            meshEdgeId = exp->GetGeom()->GetEid(j);
            int nEdgeInt;
 
            if (nFaces)
            {
                nEdgeInt =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else
            {
                nEdgeInt = exp->GetTraceNcoeffs(j) - 2;
            }
 
            if (EdgeSize.count(meshEdgeId) > 0)
            {
                EdgeSize[meshEdgeId] = min(EdgeSize[meshEdgeId], nEdgeInt);
            }
            else
            {
                EdgeSize[meshEdgeId] = nEdgeInt;
            }
        }
 
        faceCnt = 0;
        for (j = 0; j < nFaces; ++j)
        {
            meshFaceId = exp->GetGeom()->GetFid(j);
            if (FaceSize.count(meshFaceId) > 0)
            {
                FaceSize[meshFaceId] =
                    min(FaceSize[meshFaceId], exp->GetTraceIntNcoeffs(j));
            }
            else
            {
                FaceSize[meshFaceId] = exp->GetTraceIntNcoeffs(j);
            }
            FaceSize[meshFaceId] = exp->GetTraceIntNcoeffs(j);
        }
    }
 
    /// - Periodic vertices
    for (auto &pIt : periodicVerts)
    {
        meshVertId = pIt.first;
 
        // This periodic vertex is joined to a Dirichlet condition.
        if (graph[0].count(pIt.first) != 0)
        {
            for (i = 0; i < pIt.second.size(); ++i)
            {
                meshVertId2 = pIt.second[i].id;
                if (graph[0].count(meshVertId2) == 0 && pIt.second[i].isLocal)
                {
                    graph[0][meshVertId2] = graph[0][meshVertId];
                }
            }
            continue;
        }
 
        // One of the attached vertices is Dirichlet.
        bool isDirichlet = false;
        for (i = 0; i < pIt.second.size(); ++i)
        {
            if (!pIt.second[i].isLocal)
            {
                continue;
            }
 
            meshVertId2 = pIt.second[i].id;
            if (graph[0].count(meshVertId2) > 0)
            {
                isDirichlet = true;
                break;
            }
        }
 
        if (isDirichlet)
        {
            graph[0][meshVertId] = graph[0][pIt.second[i].id];
 
            for (j = 0; j < pIt.second.size(); ++j)
            {
                meshVertId2 = pIt.second[i].id;
                if (j == i || !pIt.second[j].isLocal ||
                    graph[0].count(meshVertId2) > 0)
                {
                    continue;
                }
 
                graph[0][meshVertId2] = graph[0][pIt.second[i].id];
            }
 
            continue;
        }
 
        // Otherwise, see if a vertex ID has already been set.
        for (i = 0; i < pIt.second.size(); ++i)
        {
            if (!pIt.second[i].isLocal)
            {
                continue;
            }
 
            if (tempGraph[0].count(pIt.second[i].id) > 0)
            {
                break;
            }
        }
 
        if (i == pIt.second.size())
        {
            boost::add_vertex(boostGraphObj);
            tempGraph[0][meshVertId] = tempGraphVertId++;
            m_numNonDirVertexModes++;
        }
        else
        {
            tempGraph[0][meshVertId] = tempGraph[0][pIt.second[i].id];
        }
    }
 
    // Store the temporary graph vertex id's of all element edges and
    // vertices in these 3 arrays below
    localVerts = Array<OneD, int>(nTotalVerts, -1);
    localEdges = Array<OneD, int>(nTotalEdges, -1);
    localFaces = Array<OneD, int>(nTotalFaces, -1);
 
    // Set up vertex numbering
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp     = locExpVector[i];
        vertCnt = 0;
        nVerts  = exp->GetNverts();
        for (j = 0; j < nVerts; ++j)
        {
            meshVertId = exp->GetGeom()->GetVid(j);
            if (graph[0].count(meshVertId) == 0)
            {
                if (tempGraph[0].count(meshVertId) == 0)
                {
                    boost::add_vertex(boostGraphObj);
                    tempGraph[0][meshVertId] = tempGraphVertId++;
                    m_numNonDirVertexModes += 1;
                }
                localVerts[localVertOffset + vertCnt++] =
                    tempGraph[0][meshVertId];
                vwgts_map[tempGraph[0][meshVertId]] = 1;
            }
        }
 
        localVertOffset += nVerts;
    }
 
    /// - Periodic edges
    for (auto &pIt : periodicEdges)
    {
        meshEdgeId = pIt.first;
 
        // This periodic edge is joined to a Dirichlet condition.
        if (graph[1].count(pIt.first) != 0)
        {
            for (i = 0; i < pIt.second.size(); ++i)
            {
                meshEdgeId2 = pIt.second[i].id;
                if (graph[1].count(meshEdgeId2) == 0 && pIt.second[i].isLocal)
                {
                    graph[1][meshEdgeId2] = graph[1][meshEdgeId];
                }
            }
            continue;
        }
 
        // One of the attached edges is Dirichlet.
        bool isDirichlet = false;
        for (i = 0; i < pIt.second.size(); ++i)
        {
            if (!pIt.second[i].isLocal)
            {
                continue;
            }
 
            meshEdgeId2 = pIt.second[i].id;
            if (graph[1].count(meshEdgeId2) > 0)
            {
                isDirichlet = true;
                break;
            }
        }
 
        if (isDirichlet)
        {
            graph[1][meshEdgeId] = graph[1][pIt.second[i].id];
 
            for (j = 0; j < pIt.second.size(); ++j)
            {
                meshEdgeId2 = pIt.second[i].id;
                if (j == i || !pIt.second[j].isLocal ||
                    graph[1].count(meshEdgeId2) > 0)
                {
                    continue;
                }
 
                graph[1][meshEdgeId2] = graph[1][pIt.second[i].id];
            }
 
            continue;
        }
 
        // Otherwise, see if a edge ID has already been set.
        for (i = 0; i < pIt.second.size(); ++i)
        {
            if (!pIt.second[i].isLocal)
            {
                continue;
            }
 
            if (tempGraph[1].count(pIt.second[i].id) > 0)
            {
                break;
            }
        }
 
        if (i == pIt.second.size())
        {
            boost::add_vertex(boostGraphObj);
            tempGraph[1][meshEdgeId] = tempGraphVertId++;
            m_numNonDirEdgeModes += EdgeSize[meshEdgeId];
            m_numNonDirEdges++;
        }
        else
        {
            tempGraph[1][meshEdgeId] = tempGraph[1][pIt.second[i].id];
        }
    }
 
    int nEdgeIntCoeffs, nFaceIntCoeffs;
 
    // Set up edge numbering
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp     = locExpVector[i];
        edgeCnt = 0;
        nEdges  = exp->GetGeom()->GetNumEdges();
 
        for (j = 0; j < nEdges; ++j)
        {
            meshEdgeId     = exp->GetGeom()->GetEid(j);
            nEdgeIntCoeffs = EdgeSize[meshEdgeId];
            if (graph[1].count(meshEdgeId) == 0)
            {
                if (tempGraph[1].count(meshEdgeId) == 0)
                {
                    boost::add_vertex(boostGraphObj);
                    tempGraph[1][meshEdgeId] = tempGraphVertId++;
                    m_numNonDirEdgeModes += nEdgeIntCoeffs;
 
                    m_numNonDirEdges++;
                }
                localEdges[localEdgeOffset + edgeCnt++] =
                    tempGraph[1][meshEdgeId];
                vwgts_map[tempGraph[1][meshEdgeId]] = nEdgeIntCoeffs;
            }
        }
 
        localEdgeOffset += nEdges;
    }
 
    /// - Periodic faces
    for (auto &pIt : periodicFaces)
    {
        if (!pIt.second[0].isLocal)
        {
            // The face mapped to is on another process.
            meshFaceId = pIt.first;
            ASSERTL0(graph[2].count(meshFaceId) == 0,
                     "This periodic boundary edge has been specified before");
            boost::add_vertex(boostGraphObj);
            tempGraph[2][meshFaceId] = tempGraphVertId++;
            nFaceIntCoeffs           = FaceSize[meshFaceId];
            m_numNonDirFaceModes += nFaceIntCoeffs;
            m_numNonDirFaces++;
        }
        else if (pIt.first < pIt.second[0].id)
        {
            ASSERTL0(graph[2].count(pIt.first) == 0,
                     "This periodic boundary face has been specified before");
            ASSERTL0(graph[2].count(pIt.second[0].id) == 0,
                     "This periodic boundary face has been specified before");
 
            boost::add_vertex(boostGraphObj);
            tempGraph[2][pIt.first]        = tempGraphVertId;
            tempGraph[2][pIt.second[0].id] = tempGraphVertId++;
            nFaceIntCoeffs                 = FaceSize[pIt.first];
            m_numNonDirFaceModes += nFaceIntCoeffs;
            m_numNonDirFaces++;
        }
    }
 
    // setup face numbering
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp     = locExpVector[i];
        nFaces  = exp->GetGeom()->GetNumFaces();
        faceCnt = 0;
        for (j = 0; j < nFaces; ++j)
        {
            nFaceIntCoeffs = exp->GetTraceIntNcoeffs(j);
            meshFaceId     = exp->GetGeom()->GetFid(j);
            if (graph[2].count(meshFaceId) == 0)
            {
                if (tempGraph[2].count(meshFaceId) == 0)
                {
                    boost::add_vertex(boostGraphObj);
                    tempGraph[2][meshFaceId] = tempGraphVertId++;
                    m_numNonDirFaceModes += nFaceIntCoeffs;
 
                    m_numNonDirFaces++;
                }
                localFaces[localFaceOffset + faceCnt++] =
                    tempGraph[2][meshFaceId];
                vwgts_map[tempGraph[2][meshFaceId]] = nFaceIntCoeffs;
            }
        }
        m_numLocalBndCoeffs += exp->NumBndryCoeffs();
 
        localFaceOffset += nFaces;
    }
 
    localVertOffset = 0;
    localEdgeOffset = 0;
    localFaceOffset = 0;
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp    = locExpVector[i];
        nVerts = exp->GetNverts();
        nEdges = exp->GetGeom()->GetNumEdges();
        nFaces = exp->GetGeom()->GetNumFaces();
 
        // Now loop over all local faces, edges and vertices of this
        // element and define that all other faces, edges and verices of
        // this element are adjacent to them.
 
        // Vertices
        for (j = 0; j < nVerts; j++)
        {
            if (localVerts[j + localVertOffset] == -1)
            {
                break;
            }
            // associate to other vertices
            for (k = 0; k < nVerts; k++)
            {
                if (localVerts[k + localVertOffset] == -1)
                {
                    break;
                }
                if (k != j)
                {
                    boost::add_edge((size_t)localVerts[j + localVertOffset],
                                    (size_t)localVerts[k + localVertOffset],
                                    boostGraphObj);
                }
            }
            // associate to other edges
            for (k = 0; k < nEdges; k++)
            {
                if (localEdges[k + localEdgeOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localVerts[j + localVertOffset],
                                (size_t)localEdges[k + localEdgeOffset],
                                boostGraphObj);
            }
            // associate to other faces
            for (k = 0; k < nFaces; k++)
            {
                if (localFaces[k + localFaceOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localVerts[j + localVertOffset],
                                (size_t)localFaces[k + localFaceOffset],
                                boostGraphObj);
            }
        }
 
        // Edges
        for (j = 0; j < nEdges; j++)
        {
            if (localEdges[j + localEdgeOffset] == -1)
            {
                break;
            }
            // Associate to other edges
            for (k = 0; k < nEdges; k++)
            {
                if (localEdges[k + localEdgeOffset] == -1)
                {
                    break;
                }
                if (k != j)
                {
                    boost::add_edge((size_t)localEdges[j + localEdgeOffset],
                                    (size_t)localEdges[k + localEdgeOffset],
                                    boostGraphObj);
                }
            }
            // Associate to vertices
            for (k = 0; k < nVerts; k++)
            {
                if (localVerts[k + localVertOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localEdges[j + localEdgeOffset],
                                (size_t)localVerts[k + localVertOffset],
                                boostGraphObj);
            }
            // Associate to faces
            for (k = 0; k < nFaces; k++)
            {
                if (localFaces[k + localFaceOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localEdges[j + localEdgeOffset],
                                (size_t)localFaces[k + localFaceOffset],
                                boostGraphObj);
            }
        }
 
        // Faces
        for (j = 0; j < nFaces; j++)
        {
            if (localFaces[j + localFaceOffset] == -1)
            {
                break;
            }
            // Associate to other faces
            for (k = 0; k < nFaces; k++)
            {
                if (localFaces[k + localFaceOffset] == -1)
                {
                    break;
                }
                if (k != j)
                {
                    boost::add_edge((size_t)localFaces[j + localFaceOffset],
                                    (size_t)localFaces[k + localFaceOffset],
                                    boostGraphObj);
                }
            }
            // Associate to vertices
            for (k = 0; k < nVerts; k++)
            {
                if (localVerts[k + localVertOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localFaces[j + localFaceOffset],
                                (size_t)localVerts[k + localVertOffset],
                                boostGraphObj);
            }
            // Associate to edges
            for (k = 0; k < nEdges; k++)
            {
                if (localEdges[k + localEdgeOffset] == -1)
                {
                    break;
                }
                boost::add_edge((size_t)localFaces[j + localFaceOffset],
                                (size_t)localEdges[k + localEdgeOffset],
                                boostGraphObj);
            }
        }
 
        localVertOffset += nVerts;
        localEdgeOffset += nEdges;
        localFaceOffset += nFaces;
    }
 
    // Container to store vertices of the graph which correspond to
    // degrees of freedom along the boundary and periodic BCs.
    set<int> partVerts;
 
    if (m_solnType == eIterativeMultiLevelStaticCond ||
        m_solnType == eXxtMultiLevelStaticCond)
    {
        vector<long> procVerts, procEdges, procFaces;
        set<int> foundVerts, foundEdges, foundFaces;
 
        // Loop over element and construct the procVerts and procEdges
        // vectors, which store the geometry IDs of mesh vertices and
        // edges respectively which are local to this process.
        for (i = cnt = 0; i < locExpVector.size(); ++i)
        {
            int elmtid = i;
            exp        = locExpVector[elmtid];
            for (j = 0; j < exp->GetNverts(); ++j)
            {
                int vid = exp->GetGeom()->GetVid(j) + 1;
                if (foundVerts.count(vid) == 0)
                {
                    procVerts.push_back(vid);
                    foundVerts.insert(vid);
                }
            }
 
            for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
            {
                int eid = exp->GetGeom()->GetEid(j) + 1;
 
                if (foundEdges.count(eid) == 0)
                {
                    procEdges.push_back(eid);
                    foundEdges.insert(eid);
                }
            }
 
            for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
            {
                int fid = exp->GetGeom()->GetFid(j) + 1;
 
                if (foundFaces.count(fid) == 0)
                {
                    procFaces.push_back(fid);
                    foundFaces.insert(fid);
                }
            }
        }
 
        int unique_verts = foundVerts.size();
        int unique_edges = foundEdges.size();
        int unique_faces = foundFaces.size();
 
        bool verbose = m_session->DefinesCmdLineArgument("verbose");
 
        // Now construct temporary GS objects. These will be used to
        // populate the arrays tmp3 and tmp4 with the multiplicity of
        // the vertices and edges respectively to identify those
        // vertices and edges which are located on partition boundary.
        Array<OneD, long> vertArray(unique_verts, &procVerts[0]);
        Gs::gs_data *tmp1 = Gs::Init(vertArray, vComm, verbose);
        Array<OneD, NekDouble> tmp4(unique_verts, 1.0);
        Array<OneD, NekDouble> tmp5(unique_edges, 1.0);
        Array<OneD, NekDouble> tmp6(unique_faces, 1.0);
        Gs::Gather(tmp4, Gs::gs_add, tmp1);
        Gs::Finalise(tmp1);
 
        if (unique_edges > 0)
        {
            Array<OneD, long> edgeArray(unique_edges, &procEdges[0]);
            Gs::gs_data *tmp2 = Gs::Init(edgeArray, vComm, verbose);
            Gs::Gather(tmp5, Gs::gs_add, tmp2);
            Gs::Finalise(tmp2);
        }
 
        if (unique_faces > 0)
        {
            Array<OneD, long> faceArray(unique_faces, &procFaces[0]);
            Gs::gs_data *tmp3 = Gs::Init(faceArray, vComm, verbose);
            Gs::Gather(tmp6, Gs::gs_add, tmp3);
            Gs::Finalise(tmp3);
        }
 
        // Finally, fill the partVerts set with all non-Dirichlet
        // vertices which lie on a partition boundary.
        for (i = 0; i < unique_verts; ++i)
        {
            if (tmp4[i] > 1.0)
            {
                if (graph[0].count(procVerts[i] - 1) == 0)
                {
                    partVerts.insert(tempGraph[0][procVerts[i] - 1]);
                }
            }
        }
 
        for (i = 0; i < unique_edges; ++i)
        {
            if (tmp5[i] > 1.0)
            {
                if (graph[1].count(procEdges[i] - 1) == 0)
                {
                    partVerts.insert(tempGraph[1][procEdges[i] - 1]);
                }
            }
        }
 
        for (i = 0; i < unique_faces; ++i)
        {
            if (tmp6[i] > 1.0)
            {
                if (graph[2].count(procFaces[i] - 1) == 0)
                {
                    partVerts.insert(tempGraph[2][procFaces[i] - 1]);
                }
            }
        }
 
        // Now fill with all vertices on periodic BCs
        for (auto &pIt : periodicVerts)
        {
            if (graph[0].count(pIt.first) == 0)
            {
                partVerts.insert(tempGraph[0][pIt.first]);
            }
        }
        for (auto &pIt : periodicEdges)
        {
            if (graph[1].count(pIt.first) == 0)
            {
                partVerts.insert(tempGraph[1][pIt.first]);
            }
        }
        for (auto &pIt : periodicFaces)
        {
            if (graph[2].count(pIt.first) == 0)
            {
                partVerts.insert(tempGraph[2][pIt.first]);
            }
        }
    }
 
    int nGraphVerts = tempGraphVertId;
    Array<OneD, int> perm(nGraphVerts);
    Array<OneD, int> iperm(nGraphVerts);
    Array<OneD, int> vwgts(nGraphVerts);
    ASSERTL1(vwgts_map.size() == nGraphVerts, "Non matching dimensions");
    for (i = 0; i < nGraphVerts; ++i)
    {
        vwgts[i] = vwgts_map[i];
    }
 
    if (nGraphVerts)
    {
        switch (m_solnType)
        {
            case eDirectFullMatrix:
            case eIterativeFull:
            case eIterativeStaticCond:
            case ePETScStaticCond:
            case ePETScFullMatrix:
            case eXxtFullMatrix:
            case eXxtStaticCond:
            {
                NoReordering(boostGraphObj, perm, iperm);
                break;
            }
 
            case eDirectStaticCond:
            {
                CuthillMckeeReordering(boostGraphObj, perm, iperm);
                break;
            }
 
            case ePETScMultiLevelStaticCond:
            case eDirectMultiLevelStaticCond:
            case eIterativeMultiLevelStaticCond:
            case eXxtMultiLevelStaticCond:
            {
                MultiLevelBisectionReordering(boostGraphObj, perm, iperm,
                                              bottomUpGraph, partVerts,
                                              mdswitch);
                break;
            }
            default:
            {
                ASSERTL0(false,
                         "Unrecognised solution type: " +
                             std::string(GlobalSysSolnTypeMap[m_solnType]));
            }
        }
    }
 
    // For parallel multi-level static condensation determine the lowest
    // static condensation level amongst processors.
    if ((m_solnType == eDirectMultiLevelStaticCond ||
         m_solnType == ePETScMultiLevelStaticCond ||
         m_solnType == eIterativeMultiLevelStaticCond ||
         m_solnType == eXxtMultiLevelStaticCond) &&
        bottomUpGraph)
    {
        m_lowestStaticCondLevel = bottomUpGraph->GetNlevels() - 1;
        vComm->AllReduce(m_lowestStaticCondLevel, LibUtilities::ReduceMax);
    }
    else
    {
        m_lowestStaticCondLevel = 0;
    }
 
    /**
     * STEP 4: Fill the #graph[0] and
     * #graph[1] with the optimal ordering from boost.
     */
    for (auto &mapIt : tempGraph[0])
    {
        graph[0][mapIt.first] = iperm[mapIt.second] + graphVertId;
    }
    for (auto &mapIt : tempGraph[1])
    {
        graph[1][mapIt.first] = iperm[mapIt.second] + graphVertId;
    }
    for (auto &mapIt : tempGraph[2])
    {
        graph[2][mapIt.first] = iperm[mapIt.second] + graphVertId;
    }
 
    return nGraphVerts;
}
 
/**
 *
 */
AssemblyMapCG::AssemblyMapCG(
    const LibUtilities::SessionReaderSharedPtr &pSession,
    const int numLocalCoeffs, const ExpList &locExp,
    const BndCondExp &bndCondExp, const BndCond &bndConditions,
    const bool checkIfSystemSingular, const std::string variable,
    const PeriodicMap &periodicVerts, const PeriodicMap &periodicEdges,
    const PeriodicMap &periodicFaces)
    : AssemblyMap(pSession, locExp.GetComm(), variable)
{
    int i, j, k;
    int p, q, numModes0, numModes1;
    int cnt = 0;
    int meshVertId, meshEdgeId, meshEdgeId2, meshFaceId, meshFaceId2;
    int globalId;
    int nEdgeInteriorCoeffs;
    int firstNonDirGraphVertId;
    LibUtilities::CommSharedPtr vComm = m_comm->GetRowComm();
    LocalRegions::ExpansionSharedPtr exp, bndExp;
    StdRegions::Orientation edgeOrient;
    StdRegions::Orientation faceOrient;
    Array<OneD, unsigned int> edgeInteriorMap;
    Array<OneD, int> edgeInteriorSign;
    Array<OneD, unsigned int> faceInteriorMap;
    Array<OneD, int> faceInteriorSign;
 
    const LocalRegions::ExpansionVector &locExpVector = *(locExp.GetExp());
 
    bool verbose = m_session->DefinesCmdLineArgument("verbose");
 
    m_signChange = false;
 
    // Stores vertex, edge and face reordered vertices.
    DofGraph graph(3);
    DofGraph dofs(3);
    vector<map<int, int>> faceModes(2);
    map<int, LibUtilities::ShapeType> faceType;
 
    set<int> extraDirVerts, extraDirEdges;
    BottomUpSubStructuredGraphSharedPtr bottomUpGraph;
 
    // Construct list of number of degrees of freedom for each vertex,
    // edge and face.
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
 
        for (j = 0; j < exp->GetNverts(); ++j)
        {
            dofs[0][exp->GetGeom()->GetVid(j)] = 1;
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
        {
            int nEdgeInt;
            if (exp->GetGeom()->GetNumFaces())
            {
                nEdgeInt =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else
            {
                nEdgeInt = exp->GetTraceNcoeffs(j) - 2;
            }
 
            if (dofs[1].count(exp->GetGeom()->GetEid(j)) > 0)
            {
                if (dofs[1][exp->GetGeom()->GetEid(j)] != nEdgeInt)
                {
                    ASSERTL0(
                        (exp->GetBasisType(0) == LibUtilities::eModified_A) ||
                            (exp->GetBasisType(1) ==
                             LibUtilities::eModified_B) ||
                            (exp->GetBasisType(2) ==
                             LibUtilities::eModified_C) ||
                            (exp->GetBasisType(2) ==
                             LibUtilities::eModifiedPyr_C),
                        "CG with variable order only available with "
                        "modal expansion");
                }
                dofs[1][exp->GetGeom()->GetEid(j)] =
                    min(dofs[1][exp->GetGeom()->GetEid(j)], nEdgeInt);
            }
            else
            {
                dofs[1][exp->GetGeom()->GetEid(j)] = nEdgeInt;
            }
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
        {
            faceOrient = exp->GetGeom()->GetForient(j);
            meshFaceId = exp->GetGeom()->GetFid(j);
            exp->GetTraceNumModes(j, numModes0, numModes1, faceOrient);
 
            if (faceModes[0].count(meshFaceId) > 0)
            {
                faceModes[0][meshFaceId] =
                    min(faceModes[0][meshFaceId], numModes0);
 
                faceModes[1][meshFaceId] =
                    min(faceModes[1][meshFaceId], numModes1);
            }
            else
            {
                faceModes[0][meshFaceId] = numModes0;
                faceModes[1][meshFaceId] = numModes1;
 
                // Get shape of this face
                SpatialDomains::Geometry3DSharedPtr geom;
                geom = std::dynamic_pointer_cast<SpatialDomains::Geometry3D>(
                    exp->GetGeom());
                faceType[meshFaceId] = geom->GetFace(j)->GetShapeType();
            }
        }
    }
 
    // Add non-local periodic dofs to the map
    for (auto &pIt : periodicEdges)
    {
        for (i = 0; i < pIt.second.size(); ++i)
        {
            meshEdgeId2 = pIt.second[i].id;
            if (dofs[1].count(meshEdgeId2) == 0)
            {
                dofs[1][meshEdgeId2] = 1e6;
            }
        }
    }
    for (auto &pIt : periodicFaces)
    {
        for (i = 0; i < pIt.second.size(); ++i)
        {
            meshFaceId2 = pIt.second[i].id;
            if (faceModes[0].count(meshFaceId2) == 0)
            {
                faceModes[0][meshFaceId2] = 1e6;
                faceModes[1][meshFaceId2] = 1e6;
            }
        }
    }
 
    // Now use information from all partitions to determine the correct
    // size
 
    // edges
    Array<OneD, long> edgeId(dofs[1].size());
    Array<OneD, NekDouble> edgeDof(dofs[1].size());
    i = 0;
    for (auto &dofIt : dofs[1])
    {
        edgeId[i]    = dofIt.first + 1;
        edgeDof[i++] = (NekDouble)dofIt.second;
    }
    Gs::gs_data *tmp = Gs::Init(edgeId, vComm, verbose);
    Gs::Gather(edgeDof, Gs::gs_min, tmp);
    Gs::Finalise(tmp);
    for (i = 0; i < dofs[1].size(); i++)
    {
        dofs[1][edgeId[i] - 1] = (int)(edgeDof[i] + 0.5);
    }
    // Periodic edges
    for (auto &pIt : periodicEdges)
    {
        meshEdgeId = pIt.first;
        for (i = 0; i < pIt.second.size(); ++i)
        {
            meshEdgeId2 = pIt.second[i].id;
            if (dofs[1][meshEdgeId2] < dofs[1][meshEdgeId])
            {
                dofs[1][meshEdgeId] = dofs[1][meshEdgeId2];
            }
        }
    }
    // faces
    Array<OneD, long> faceId(faceModes[0].size());
    Array<OneD, NekDouble> faceP(faceModes[0].size());
    Array<OneD, NekDouble> faceQ(faceModes[0].size());
 
    i = 0;
    for (auto dofIt = faceModes[0].begin(), dofIt2 = faceModes[1].begin();
         dofIt != faceModes[0].end(); dofIt++, dofIt2++, i++)
    {
        faceId[i] = dofIt->first + 1;
        faceP[i]  = (NekDouble)dofIt->second;
        faceQ[i]  = (NekDouble)dofIt2->second;
    }
    Gs::gs_data *tmp2 = Gs::Init(faceId, vComm, verbose);
    Gs::Gather(faceP, Gs::gs_min, tmp2);
    Gs::Gather(faceQ, Gs::gs_min, tmp2);
    Gs::Finalise(tmp2);
    for (i = 0; i < faceModes[0].size(); i++)
    {
        faceModes[0][faceId[i] - 1] = (int)(faceP[i] + 0.5);
        faceModes[1][faceId[i] - 1] = (int)(faceQ[i] + 0.5);
    }
    // Periodic faces
    for (auto &pIt : periodicFaces)
    {
        meshFaceId = pIt.first;
        for (i = 0; i < pIt.second.size(); ++i)
        {
            meshFaceId2 = pIt.second[i].id;
            if (faceModes[0][meshFaceId2] < faceModes[0][meshFaceId])
            {
                faceModes[0][meshFaceId] = faceModes[0][meshFaceId2];
            }
            if (faceModes[1][meshFaceId2] < faceModes[1][meshFaceId])
            {
                faceModes[1][meshFaceId] = faceModes[1][meshFaceId2];
            }
        }
    }
    // Calculate number of dof in each face
    int P, Q;
    for (i = 0; i < faceModes[0].size(); i++)
    {
        P = faceModes[0][faceId[i] - 1];
        Q = faceModes[1][faceId[i] - 1];
        if (faceType[faceId[i] - 1] == LibUtilities::eQuadrilateral)
        {
            // Quad face
            dofs[2][faceId[i] - 1] =
                LibUtilities::StdQuadData::getNumberOfCoefficients(P, Q) -
                LibUtilities::StdQuadData::getNumberOfBndCoefficients(P, Q);
        }
        else
        {
            // Tri face
            dofs[2][faceId[i] - 1] =
                LibUtilities::StdTriData::getNumberOfCoefficients(P, Q) -
                LibUtilities::StdTriData::getNumberOfBndCoefficients(P, Q);
        }
    }
 
    Array<OneD, const BndCond> bndCondVec(1, bndConditions);
 
    // Note that nExtraDirichlet is not used in the logic below; it just
    // needs to be set so that the coupled solver in
    // IncNavierStokesSolver can work.
    int nExtraDirichlet;
    int mdswitch;
    m_session->LoadParameter("MDSwitch", mdswitch, 10);
 
    int nGraphVerts = CreateGraph(
        locExp, bndCondExp, bndCondVec, checkIfSystemSingular, periodicVerts,
        periodicEdges, periodicFaces, graph, bottomUpGraph, extraDirVerts,
        extraDirEdges, firstNonDirGraphVertId, nExtraDirichlet, mdswitch);
 
    /*
     * Set up an array which contains the offset information of the
     * different graph vertices.
     *
     * This basically means to identify to how many global degrees of
     * freedom the individual graph vertices correspond. Obviously,
     * the graph vertices corresponding to the mesh-vertices account
     * for a single global DOF. However, the graph vertices
     * corresponding to the element edges correspond to N-2 global DOF
     * where N is equal to the number of boundary modes on this edge.
     */
    Array<OneD, int> graphVertOffset(
        graph[0].size() + graph[1].size() + graph[2].size() + 1, 0);
 
    graphVertOffset[0] = 0;
 
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
 
        for (j = 0; j < exp->GetNverts(); ++j)
        {
            meshVertId = exp->GetGeom()->GetVid(j);
            graphVertOffset[graph[0][meshVertId] + 1] = 1;
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
        {
            if (exp->GetGeom()->GetNumFaces()) // 3D version
            {
                nEdgeInteriorCoeffs =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else
            {
                nEdgeInteriorCoeffs = exp->GetTraceNcoeffs(j) - 2;
            }
            meshEdgeId = exp->GetGeom()->GetEid(j);
            graphVertOffset[graph[1][meshEdgeId] + 1] = dofs[1][meshEdgeId];
 
            // Need a sign vector for modal expansions if nEdgeCoeffs
            // >=3 (not 4 because of variable order case)
            if (nEdgeInteriorCoeffs &&
                (exp->GetBasisType(0) == LibUtilities::eModified_A))
            {
                m_signChange = true;
            }
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
        {
            meshFaceId = exp->GetGeom()->GetFid(j);
            graphVertOffset[graph[2][meshFaceId] + 1] = dofs[2][meshFaceId];
        }
    }
 
    for (i = 1; i < graphVertOffset.size(); i++)
    {
        graphVertOffset[i] += graphVertOffset[i - 1];
    }
 
    // Allocate the proper amount of space for the class-data
    m_numLocalCoeffs        = numLocalCoeffs;
    m_numGlobalDirBndCoeffs = graphVertOffset[firstNonDirGraphVertId];
    m_localToGlobalMap      = Array<OneD, int>(m_numLocalCoeffs, -1);
    m_localToGlobalBndMap   = Array<OneD, int>(m_numLocalBndCoeffs, -1);
    m_localToLocalBndMap    = Array<OneD, int>(m_numLocalBndCoeffs, -1);
    m_localToLocalIntMap =
        Array<OneD, int>(m_numLocalCoeffs - m_numLocalBndCoeffs, -1);
    m_bndCondCoeffsToLocalCoeffsMap =
        Array<OneD, int>(m_numLocalBndCondCoeffs, -1);
 
    // If required, set up the sign-vector
    if (m_signChange)
    {
        m_localToGlobalSign = Array<OneD, NekDouble>(m_numLocalCoeffs, 1.0);
        m_localToGlobalBndSign =
            Array<OneD, NekDouble>(m_numLocalBndCoeffs, 1.0);
        m_bndCondCoeffsToLocalCoeffsSign =
            Array<OneD, NekDouble>(m_numLocalBndCondCoeffs, 1.0);
    }
 
    m_staticCondLevel           = 0;
    m_numPatches                = locExpVector.size();
    m_numLocalBndCoeffsPerPatch = Array<OneD, unsigned int>(m_numPatches);
    m_numLocalIntCoeffsPerPatch = Array<OneD, unsigned int>(m_numPatches);
    for (i = 0; i < m_numPatches; ++i)
    {
        m_numLocalBndCoeffsPerPatch[i] =
            (unsigned int)locExpVector[i]->NumBndryCoeffs();
        m_numLocalIntCoeffsPerPatch[i] =
            (unsigned int)locExpVector[i]->GetNcoeffs() -
            locExpVector[i]->NumBndryCoeffs();
    }
 
    /**
     * STEP 6: Now, all ingredients are ready to set up the actual
     * local to global mapping.
     *
     * The remainder of the map consists of the element-interior
     * degrees of freedom. This leads to the block-diagonal submatrix
     * as each element-interior mode is globally orthogonal to modes
     * in all other elements.
     */
    cnt = 0;
 
    // Loop over all the elements in the domain
    int cntbdry = 0;
    int cntint  = 0;
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
        cnt = locExp.GetCoeff_Offset(i);
 
        int nbdry = exp->NumBndryCoeffs();
        int nint  = exp->GetNcoeffs() - nbdry;
 
        Array<OneD, unsigned int> bmap(nbdry);
        Array<OneD, unsigned int> imap(nint);
 
        exp->GetBoundaryMap(bmap);
        exp->GetInteriorMap(imap);
 
        for (j = 0; j < nbdry; ++j)
        {
            m_localToLocalBndMap[cntbdry++] = cnt + bmap[j];
        }
 
        for (j = 0; j < nint; ++j)
        {
            m_localToLocalIntMap[cntint++] = cnt + imap[j];
        }
 
        for (j = 0; j < exp->GetNverts(); ++j)
        {
            meshVertId = exp->GetGeom()->GetVid(j);
 
            // Set the global DOF for vertex j of element i
            m_localToGlobalMap[cnt + exp->GetVertexMap(j)] =
                graphVertOffset[graph[0][meshVertId]];
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
        {
            if (exp->GetGeom()->GetNumFaces())
            {
                nEdgeInteriorCoeffs =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else
            {
                nEdgeInteriorCoeffs = exp->GetTraceNcoeffs(j) - 2;
            }
            edgeOrient = exp->GetGeom()->GetEorient(j);
            meshEdgeId = exp->GetGeom()->GetEid(j);
 
            auto pIt = periodicEdges.find(meshEdgeId);
 
            // See if this edge is periodic. If it is, then we map all
            // edges to the one with lowest ID, and align all
            // coefficients to this edge orientation.
            if (pIt != periodicEdges.end())
            {
                pair<int, StdRegions::Orientation> idOrient =
                    DeterminePeriodicEdgeOrientId(meshEdgeId, edgeOrient,
                                                  pIt->second);
                edgeOrient = idOrient.second;
            }
 
            if (exp->GetGeom()->GetNumFaces())
            {
                exp->as<LocalRegions::Expansion3D>()
                    ->GetEdgeInteriorToElementMap(j, edgeInteriorMap,
                                                  edgeInteriorSign, edgeOrient);
            }
            else
            {
                exp->GetTraceInteriorToElementMap(j, edgeInteriorMap,
                                                  edgeInteriorSign, edgeOrient);
            }
 
            // Set the global DOF's for the interior modes of edge j
            for (k = 0; k < dofs[1][meshEdgeId]; ++k)
            {
                m_localToGlobalMap[cnt + edgeInteriorMap[k]] =
                    graphVertOffset[graph[1][meshEdgeId]] + k;
            }
            for (k = dofs[1][meshEdgeId]; k < nEdgeInteriorCoeffs; ++k)
            {
                m_localToGlobalMap[cnt + edgeInteriorMap[k]] = 0;
            }
 
            // Fill the sign vector if required
            if (m_signChange)
            {
                for (k = 0; k < dofs[1][meshEdgeId]; ++k)
                {
                    m_localToGlobalSign[cnt + edgeInteriorMap[k]] =
                        (NekDouble)edgeInteriorSign[k];
                }
                for (k = dofs[1][meshEdgeId]; k < nEdgeInteriorCoeffs; ++k)
                {
                    m_localToGlobalSign[cnt + edgeInteriorMap[k]] = 0.0;
                }
            }
        }
 
        for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
        {
            faceOrient = exp->GetGeom()->GetForient(j);
            meshFaceId = exp->GetGeom()->GetFid(j);
 
            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(j, faceInteriorMap,
                                              faceInteriorSign, faceOrient);
 
            // Set the global DOF's for the interior modes of face j
            exp->GetTraceNumModes(j, numModes0, numModes1, faceOrient);
            switch (faceType[meshFaceId])
            {
                case LibUtilities::eQuadrilateral:
                {
                    int kLoc = 0;
                    k        = 0;
                    for (q = 2; q < numModes1; q++)
                    {
                        for (p = 2; p < numModes0; p++)
                        {
                            if ((p < faceModes[0][meshFaceId]) &&
                                (q < faceModes[1][meshFaceId]))
                            {
                                m_localToGlobalMap[cnt +
                                                   faceInteriorMap[kLoc]] =
                                    graphVertOffset[graph[2][meshFaceId]] + k;
                                if (m_signChange)
                                {
                                    m_localToGlobalSign[cnt +
                                                        faceInteriorMap[kLoc]] =
                                        (NekDouble)faceInteriorSign[kLoc];
                                }
                                k++;
                            }
                            else
                            {
                                m_localToGlobalMap[cnt +
                                                   faceInteriorMap[kLoc]] = 0;
                                if (m_signChange)
                                {
                                    m_localToGlobalSign[cnt +
                                                        faceInteriorMap[kLoc]] =
                                        0.0;
                                }
                            }
                            kLoc++;
                        }
                    }
                }
                break;
                case LibUtilities::eTriangle:
                {
                    int kLoc = 0;
                    k        = 0;
                    for (p = 2; p < numModes0; p++)
                    {
                        for (q = 1; q < numModes1 - p; q++)
                        {
                            if ((p < faceModes[0][meshFaceId]) &&
                                (p + q < faceModes[1][meshFaceId]))
                            {
                                m_localToGlobalMap[cnt +
                                                   faceInteriorMap[kLoc]] =
                                    graphVertOffset[graph[2][meshFaceId]] + k;
                                if (m_signChange)
                                {
                                    m_localToGlobalSign[cnt +
                                                        faceInteriorMap[kLoc]] =
                                        (NekDouble)faceInteriorSign[kLoc];
                                }
                                k++;
                            }
                            else
                            {
                                m_localToGlobalMap[cnt +
                                                   faceInteriorMap[kLoc]] = 0;
                                if (m_signChange)
                                {
                                    m_localToGlobalSign[cnt +
                                                        faceInteriorMap[kLoc]] =
                                        0.0;
                                }
                            }
                            kLoc++;
                        }
                    }
                }
                break;
                default:
                    ASSERTL0(false, "Shape not recognised");
                    break;
            }
        }
    }
 
    // Set up the mapping for the boundary conditions
    // Set up boundary mapping
    map<int, pair<int, int>> traceToElmtTraceMap;
    int id;
 
    for (cnt = i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
 
        for (j = 0; j < exp->GetNtraces(); ++j)
        {
            id = exp->GetGeom()->GetTid(j);
 
            traceToElmtTraceMap[id] = pair<int, int>(i, j);
        }
    }
 
    Array<OneD, unsigned int> maparray;
    Array<OneD, int> signarray;
    map<int, pair<int, NekDouble>> GloDirBndCoeffToLocalCoeff;
    set<int> CoeffOnDirTrace;
 
    cnt        = 0;
    int offset = 0;
    for (i = 0; i < bndCondExp.size(); i++)
    {
        set<int> foundExtraVerts, foundExtraEdges;
        for (j = 0; j < bndCondExp[i]->GetNumElmts(); j++)
        {
            bndExp = bndCondExp[i]->GetExp(j);
            cnt    = offset + bndCondExp[i]->GetCoeff_Offset(j);
 
            int id = bndExp->GetGeom()->GetGlobalID();
 
            ASSERTL1(traceToElmtTraceMap.count(id) > 0,
                     "Failed to find trace id");
 
            int eid = traceToElmtTraceMap[id].first;
            int tid = traceToElmtTraceMap[id].second;
 
            exp     = locExpVector[eid];
            int dim = exp->GetShapeDimension();
 
            if (dim == 1)
            {
                m_bndCondCoeffsToLocalCoeffsMap[cnt] =
                    locExp.GetCoeff_Offset(eid) + exp->GetVertexMap(tid);
            }
            else
            {
                if (dim == 2)
                {
                    exp->GetTraceToElementMap(tid, maparray, signarray,
                                              exp->GetGeom()->GetEorient(tid),
                                              bndExp->GetBasisNumModes(0));
                }
                else if (dim == 3)
                {
                    exp->GetTraceToElementMap(tid, maparray, signarray,
                                              exp->GetGeom()->GetForient(tid),
                                              bndExp->GetBasisNumModes(0),
                                              bndExp->GetBasisNumModes(1));
                }
 
                for (k = 0; k < bndExp->GetNcoeffs(); k++)
                {
                    m_bndCondCoeffsToLocalCoeffsMap[cnt + k] =
                        locExp.GetCoeff_Offset(eid) + maparray[k];
                    if (m_signChange)
                    {
                        m_bndCondCoeffsToLocalCoeffsSign[cnt + k] =
                            signarray[k];
                    }
                }
            }
 
            // we now need some information to work out how to
            // handle vertices and edges that are only just
            // touching a dirichlet boundary (and not the
            // whole edge/face)
 
            for (k = 0; k < bndExp->GetNcoeffs(); k++)
            {
                int locid      = m_bndCondCoeffsToLocalCoeffsMap[cnt + k];
                int gloid      = m_localToGlobalMap[locid];
                NekDouble sign = 1.0;
 
                if (m_signChange)
                {
                    sign = m_bndCondCoeffsToLocalCoeffsSign[cnt + k];
                }
 
                if (bndConditions[i]->GetBoundaryConditionType() ==
                    SpatialDomains::eDirichlet)
                {
                    CoeffOnDirTrace.insert(locid);
 
                    // store the local id and sign from global id
                    // back to local space;
                    GloDirBndCoeffToLocalCoeff[gloid] =
                        pair<int, NekDouble>(locid, sign);
                }
            }
        }
        offset += bndCondExp[i]->GetNcoeffs();
    }
 
    globalId = Vmath::Vmax(m_numLocalCoeffs, &m_localToGlobalMap[0], 1) + 1;
    m_numGlobalBndCoeffs = globalId;
 
    // Set up a mapping list of Dirichlet Local Dofs that
    // arise due to one vertex or edge just touching a
    // Dirichlet boundary and need the value from another
    // local coeff that has been filled by the boundary
    // coeffs.
 
    Array<OneD, NekDouble> gloParaDirBnd(m_numGlobalBndCoeffs, -1.0);
 
    Array<OneD, unsigned int> bndmap;
    cnt = 0;
    for (i = 0; i < locExpVector.size(); ++i)
    {
        int gloid;
 
        exp = locExpVector[i];
 
        exp->GetBoundaryMap(bndmap);
 
        for (j = 0; j < bndmap.size(); ++j)
        {
            k = cnt + bndmap[j];
 
            if (CoeffOnDirTrace.count(k) == 0)
            {
                gloid = m_localToGlobalMap[k];
 
                if (gloid < m_numGlobalDirBndCoeffs) // point on Dir BC
                {
                    if (GloDirBndCoeffToLocalCoeff.count(gloid))
                    {
                        int locid = GloDirBndCoeffToLocalCoeff[gloid].first;
                        NekDouble sign = 1.0;
 
                        if (m_signChange)
                        {
                            sign = m_localToGlobalSign[locid] *
                                   m_localToGlobalSign[k];
                        }
 
                        ExtraDirDof DirDofs(k, locid, sign);
                        // could make same `structure as extraDirDof
                        m_copyLocalDirDofs.insert(DirDofs);
                    }
                    else // else could be on another parallel partition.
                    {
                        gloParaDirBnd[gloid] = gloid;
                    }
                }
            }
        }
 
        cnt += exp->GetNcoeffs();
    }
 
    /*
     * The boundary condition mapping is generated from the same vertex
     * renumbering.
     */
    cnt = 0;
    for (i = 0; i < m_numLocalCoeffs; ++i)
    {
        if (m_localToGlobalMap[i] == -1)
        {
            m_localToGlobalMap[i] = globalId++;
        }
        else
        {
            if (m_signChange)
            {
                m_localToGlobalBndSign[cnt] = m_localToGlobalSign[i];
            }
            m_localToGlobalBndMap[cnt++] = m_localToGlobalMap[i];
        }
    }
 
    m_numGlobalCoeffs = globalId;
 
    SetUpUniversalC0ContMap(locExp, periodicVerts, periodicEdges,
                            periodicFaces);
 
    // Now that universal map is setup reset gloParaDirBnd to
    // 0 if no point communicated or universal value of not
    // equal to -1.0
    for (i = 0; i < m_numGlobalBndCoeffs; ++i)
    {
        int gloid = gloParaDirBnd[i];
        if (gloid == -1)
        {
            gloParaDirBnd[i] = 0.0;
        }
        else
        {
            gloParaDirBnd[i] = m_globalToUniversalMap[gloid];
        }
    }
 
    // Use parallel boundary communication to set parallel
    // dirichlet values on all processors Needs to be after
    // SetupUuniversialC0ContMap
    Gs::Gather(gloParaDirBnd, Gs::gs_max, m_bndGsh);
 
    // copy global ids back to local values in partition to
    // initialise gs communicator.
    Array<OneD, long> paraDirBnd(m_numLocalCoeffs);
    for (i = 0; i < numLocalCoeffs; ++i)
    {
        paraDirBnd[i] = 0.0;
 
        int id = m_localToGlobalMap[i];
 
        if (id >= m_numGlobalDirBndCoeffs)
        {
            continue;
        }
 
        paraDirBnd[i] = gloParaDirBnd[id];
 
        if (gloParaDirBnd[id] > 0.0)
        {
            // gather any sign changes due to edge modes
            if (m_signChange)
            {
                if (m_localToGlobalSign[i] < 0)
                {
                    m_parallelDirBndSign.insert(i);
                }
            }
        }
    }
 
    m_dirBndGsh = Gs::Init(paraDirBnd, vComm, verbose);
 
    // Set up the local to global map for the next level when using
    // multi-level static condensation
    if ((m_solnType == eDirectMultiLevelStaticCond ||
         m_solnType == eIterativeMultiLevelStaticCond ||
         m_solnType == eXxtMultiLevelStaticCond ||
         m_solnType == ePETScMultiLevelStaticCond) &&
        nGraphVerts)
    {
        if (m_staticCondLevel < (bottomUpGraph->GetNlevels() - 1))
        {
            Array<OneD, int> vwgts_perm(graph[0].size() + graph[1].size() +
                                        graph[2].size() -
                                        firstNonDirGraphVertId);
 
            for (i = 0; i < locExpVector.size(); ++i)
            {
                exp = locExpVector[i];
 
                for (j = 0; j < exp->GetNverts(); ++j)
                {
                    meshVertId = exp->GetGeom()->GetVid(j);
 
                    if (graph[0][meshVertId] >= firstNonDirGraphVertId)
                    {
                        vwgts_perm[graph[0][meshVertId] -
                                   firstNonDirGraphVertId] =
                            dofs[0][meshVertId];
                    }
                }
 
                for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
                {
                    meshEdgeId = exp->GetGeom()->GetEid(j);
 
                    if (graph[1][meshEdgeId] >= firstNonDirGraphVertId)
                    {
                        vwgts_perm[graph[1][meshEdgeId] -
                                   firstNonDirGraphVertId] =
                            dofs[1][meshEdgeId];
                    }
                }
 
                for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
                {
                    meshFaceId = exp->GetGeom()->GetFid(j);
 
                    if (graph[2][meshFaceId] >= firstNonDirGraphVertId)
                    {
                        vwgts_perm[graph[2][meshFaceId] -
                                   firstNonDirGraphVertId] =
                            dofs[2][meshFaceId];
                    }
                }
            }
 
            bottomUpGraph->ExpandGraphWithVertexWeights(vwgts_perm);
            m_nextLevelLocalToGlobalMap =
                MemoryManager<AssemblyMap>::AllocateSharedPtr(this,
                                                              bottomUpGraph);
        }
    }
 
    m_hash = hash_range(m_localToGlobalMap.begin(), m_localToGlobalMap.end());
 
    // Add up hash values if parallel
    int hash = m_hash;
    vComm->AllReduce(hash, LibUtilities::ReduceSum);
    m_hash = hash;
 
    CalculateBndSystemBandWidth();
    CalculateFullSystemBandWidth();
}
 
/**
 *
 */
AssemblyMapCG::~AssemblyMapCG()
{
    Gs::Finalise(m_gsh);
    Gs::Finalise(m_bndGsh);
}
 
/**
 * @brief Determine orientation of an edge to its periodic equivalents,
 * as well as the ID of the representative edge.
 *
 * Since an edge may be periodic with more than one other edge (e.g. a
 * periodic cube has sets of four periodic edges in each coordinate
 * direction), we have to define a 'representative' edge. In this
 * assembly map we define it to be the one with the minimum ID. This
 * routine is set up to calculate the orientation of a given edge with
 * ID @p meshEdgeId with respect to the edge ID.
 *
 * @param meshEdgeId     ID of a periodic edge.
 * @param edgeOrient     Edge orientation of meshEdgeId with respect to
 *                       its parent element.
 * @param periodicEdges  The map of all periodic edges.
 *
 * @return Pair containing the ID of the periodic edge and the
 *         orientation of @p meshEdgeID with respect to this edge.
 */
pair<int, StdRegions::Orientation> DeterminePeriodicEdgeOrientId(
    int meshEdgeId, StdRegions::Orientation edgeOrient,
    const vector<PeriodicEntity> &periodicEdges)
{
    int minId  = periodicEdges[0].id;
    int minIdK = 0;
    int k;
 
    for (k = 1; k < periodicEdges.size(); ++k)
    {
        if (periodicEdges[k].id < minId)
        {
            minId  = min(minId, periodicEdges[k].id);
            minIdK = k;
        }
    }
 
    minId = min(minId, meshEdgeId);
 
    if (meshEdgeId != minId)
    {
        if (periodicEdges[minIdK].orient == StdRegions::eBackwards)
        {
            // Swap edge orientation
            edgeOrient = (edgeOrient == StdRegions::eForwards)
                             ? StdRegions::eBackwards
                             : StdRegions::eForwards;
        }
    }
 
    return make_pair(minId, edgeOrient);
}
 
/**
 * @brief Determine relative orientation between two faces.
 *
 * Given the orientation of a local element to its local face, defined
 * as @p faceOrient, and @p perFaceOrient which states the alignment of
 * one periodic face to the other global face, this routine determines
 * the orientation that takes this local element face to the
 * global/unique face.
 *
 * @param faceOrient     Orientation of the face with respect to its
 *                       parent element.
 * @param perFaceOrient  Orientation of the representative/global face.
 *
 * @return Orientation between the two faces.
 */
StdRegions::Orientation DeterminePeriodicFaceOrient(
    StdRegions::Orientation faceOrient, StdRegions::Orientation perFaceOrient)
{
    StdRegions::Orientation returnval = faceOrient;
 
    if (perFaceOrient != StdRegions::eDir1FwdDir1_Dir2FwdDir2)
    {
        int tmp1 = (int)faceOrient - 5;
        int tmp2 = (int)perFaceOrient - 5;
 
        int flipDir1Map[8]  = {2, 3, 0, 1, 6, 7, 4, 5};
        int flipDir2Map[8]  = {1, 0, 3, 2, 5, 4, 7, 6};
        int transposeMap[8] = {4, 5, 6, 7, 0, 2, 1, 3};
 
        // Transpose orientation
        if (tmp2 > 3)
        {
            tmp1 = transposeMap[tmp1];
        }
 
        // Reverse orientation in direction 1.
        if (tmp2 == 2 || tmp2 == 3 || tmp2 == 6 || tmp2 == 7)
        {
            tmp1 = flipDir1Map[tmp1];
        }
 
        // Reverse orientation in direction 2
        if (tmp2 % 2 == 1)
        {
            tmp1 = flipDir2Map[tmp1];
        }
 
        returnval = (StdRegions::Orientation)(tmp1 + 5);
    }
    return returnval;
}
 
/**
 * Sets up the global to universal mapping of degrees of freedom across
 * processors.
 */
void AssemblyMapCG::SetUpUniversalC0ContMap(const ExpList &locExp,
                                            const PeriodicMap &perVerts,
                                            const PeriodicMap &perEdges,
                                            const PeriodicMap &perFaces)
{
    LocalRegions::ExpansionSharedPtr exp;
    int nVert      = 0;
    int nEdge      = 0;
    int nFace      = 0;
    int maxEdgeDof = 0;
    int maxFaceDof = 0;
    int maxIntDof  = 0;
    int dof        = 0;
    int cnt;
    int i, j, k, l;
    int meshVertId;
    int meshEdgeId;
    int meshFaceId;
    int elementId;
    int vGlobalId;
    int maxBndGlobalId = 0;
    StdRegions::Orientation edgeOrient;
    StdRegions::Orientation faceOrient;
    Array<OneD, unsigned int> edgeInteriorMap;
    Array<OneD, int> edgeInteriorSign;
    Array<OneD, unsigned int> faceInteriorMap;
    Array<OneD, int> faceInteriorSign;
    Array<OneD, unsigned int> interiorMap;
 
    const LocalRegions::ExpansionVector &locExpVector = *(locExp.GetExp());
    LibUtilities::CommSharedPtr vCommRow              = m_comm->GetRowComm();
    const bool verbose = locExp.GetSession()->DefinesCmdLineArgument("verbose");
 
    m_globalToUniversalMap = Nektar::Array<OneD, int>(m_numGlobalCoeffs, -1);
    m_globalToUniversalMapUnique =
        Nektar::Array<OneD, int>(m_numGlobalCoeffs, -1);
    m_globalToUniversalBndMap =
        Nektar::Array<OneD, int>(m_numGlobalBndCoeffs, -1);
    m_globalToUniversalBndMapUnique =
        Nektar::Array<OneD, int>(m_numGlobalBndCoeffs, -1);
 
    // Loop over all the elements in the domain to gather mesh data
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
 
        int nv = exp->GetNverts();
        int ne = exp->GetGeom()->GetNumEdges();
        int nf = exp->GetGeom()->GetNumFaces();
 
        nVert += nv;
        nEdge += ne;
        nFace += nf;
 
        // Loop over all edges (and vertices) of element i
        for (j = 0; j < ne; ++j)
        {
            if (nf)
            {
                dof =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else
            {
                dof = exp->GetTraceNcoeffs(j) - 2;
            }
 
            maxEdgeDof = (dof > maxEdgeDof ? dof : maxEdgeDof);
        }
        for (j = 0; j < nf; ++j)
        {
            dof        = exp->GetTraceIntNcoeffs(j);
            maxFaceDof = (dof > maxFaceDof ? dof : maxFaceDof);
        }
        exp->GetInteriorMap(interiorMap);
        dof       = interiorMap.size();
        maxIntDof = (dof > maxIntDof ? dof : maxIntDof);
    }
 
    // Tell other processes about how many dof we have
    vCommRow->AllReduce(nVert, LibUtilities::ReduceSum);
    vCommRow->AllReduce(nEdge, LibUtilities::ReduceSum);
    vCommRow->AllReduce(nFace, LibUtilities::ReduceSum);
    vCommRow->AllReduce(maxEdgeDof, LibUtilities::ReduceMax);
    vCommRow->AllReduce(maxFaceDof, LibUtilities::ReduceMax);
    vCommRow->AllReduce(maxIntDof, LibUtilities::ReduceMax);
 
    // Assemble global to universal mapping for this process
    for (i = 0; i < locExpVector.size(); ++i)
    {
        exp = locExpVector[i];
        cnt = locExp.GetCoeff_Offset(i);
 
        int nf = exp->GetGeom()->GetNumFaces();
 
        // Loop over all vertices of element i
        for (j = 0; j < exp->GetNverts(); ++j)
        {
            meshVertId = exp->GetGeom()->GetVid(j);
            vGlobalId  = m_localToGlobalMap[cnt + exp->GetVertexMap(j)];
 
            auto pIt = perVerts.find(meshVertId);
            if (pIt != perVerts.end())
            {
                for (k = 0; k < pIt->second.size(); ++k)
                {
                    meshVertId = min(meshVertId, pIt->second[k].id);
                }
            }
 
            m_globalToUniversalMap[vGlobalId] = meshVertId + 1;
            m_globalToUniversalBndMap[vGlobalId] =
                m_globalToUniversalMap[vGlobalId];
            maxBndGlobalId =
                (vGlobalId > maxBndGlobalId ? vGlobalId : maxBndGlobalId);
        }
 
        // Loop over all edges of element i
        for (j = 0; j < exp->GetGeom()->GetNumEdges(); ++j)
        {
            meshEdgeId = exp->GetGeom()->GetEid(j);
            auto pIt   = perEdges.find(meshEdgeId);
            edgeOrient = exp->GetGeom()->GetEorient(j);
 
            if (pIt != perEdges.end())
            {
                pair<int, StdRegions::Orientation> idOrient =
                    DeterminePeriodicEdgeOrientId(meshEdgeId, edgeOrient,
                                                  pIt->second);
                meshEdgeId = idOrient.first;
                edgeOrient = idOrient.second;
            }
 
            if (nf) // 3D version
            {
                exp->as<LocalRegions::Expansion3D>()
                    ->GetEdgeInteriorToElementMap(j, edgeInteriorMap,
                                                  edgeInteriorSign, edgeOrient);
                dof =
                    exp->as<LocalRegions::Expansion3D>()->GetEdgeNcoeffs(j) - 2;
            }
            else // 2D version
            {
                exp->GetTraceInteriorToElementMap(j, edgeInteriorMap,
                                                  edgeInteriorSign, edgeOrient);
                dof = exp->GetTraceNcoeffs(j) - 2;
            }
 
            // Set the global DOF's for the interior modes of edge j
            //    for varP, ignore modes with sign == 0
            for (k = 0, l = 0; k < dof; ++k)
            {
                if (m_signChange)
                {
                    if (m_localToGlobalSign[cnt + edgeInteriorMap[k]] == 0)
                    {
                        continue;
                    }
                }
                vGlobalId = m_localToGlobalMap[cnt + edgeInteriorMap[k]];
                m_globalToUniversalMap[vGlobalId] =
                    nVert + meshEdgeId * maxEdgeDof + l + 1;
                m_globalToUniversalBndMap[vGlobalId] =
                    m_globalToUniversalMap[vGlobalId];
                maxBndGlobalId =
                    (vGlobalId > maxBndGlobalId ? vGlobalId : maxBndGlobalId);
                l++;
            }
        }
 
        // Loop over all faces of element i
        for (j = 0; j < exp->GetGeom()->GetNumFaces(); ++j)
        {
            faceOrient = exp->GetGeom()->GetForient(j);
 
            meshFaceId = exp->GetGeom()->GetFid(j);
 
            auto pIt = perFaces.find(meshFaceId);
            if (pIt != perFaces.end())
            {
                if (meshFaceId == min(meshFaceId, pIt->second[0].id))
                {
                    faceOrient = DeterminePeriodicFaceOrient(
                        faceOrient, pIt->second[0].orient);
                }
                meshFaceId = min(meshFaceId, pIt->second[0].id);
            }
 
            exp->GetTraceInteriorToElementMap(j, faceInteriorMap,
                                              faceInteriorSign, faceOrient);
            dof = exp->GetTraceIntNcoeffs(j);
 
            for (k = 0, l = 0; k < dof; ++k)
            {
                if (m_signChange)
                {
                    if (m_localToGlobalSign[cnt + faceInteriorMap[k]] == 0)
                    {
                        continue;
                    }
                }
                vGlobalId = m_localToGlobalMap[cnt + faceInteriorMap[k]];
                m_globalToUniversalMap[vGlobalId] = nVert + nEdge * maxEdgeDof +
                                                    meshFaceId * maxFaceDof +
                                                    l + 1;
                m_globalToUniversalBndMap[vGlobalId] =
                    m_globalToUniversalMap[vGlobalId];
 
                maxBndGlobalId =
                    (vGlobalId > maxBndGlobalId ? vGlobalId : maxBndGlobalId);
                l++;
            }
        }
 
        // Add interior DOFs to complete universal numbering
        exp->GetInteriorMap(interiorMap);
        dof       = interiorMap.size();
        elementId = (exp->GetGeom())->GetGlobalID();
        for (k = 0; k < dof; ++k)
        {
            vGlobalId = m_localToGlobalMap[cnt + interiorMap[k]];
            m_globalToUniversalMap[vGlobalId] = nVert + nEdge * maxEdgeDof +
                                                nFace * maxFaceDof +
                                                elementId * maxIntDof + k + 1;
        }
    }
 
    // Set up the GSLib universal assemble mapping
    // Internal DOF do not participate in any data
    // exchange, so we keep these set to the special GSLib id=0 so
    // they are ignored.
    Nektar::Array<OneD, long> tmp(m_numGlobalCoeffs);
    Vmath::Zero(m_numGlobalCoeffs, tmp, 1);
    Nektar::Array<OneD, long> tmp2(m_numGlobalBndCoeffs, tmp);
    for (unsigned int i = 0; i < m_numGlobalBndCoeffs; ++i)
    {
        tmp[i] = m_globalToUniversalMap[i];
    }
 
    m_gsh    = Gs::Init(tmp, vCommRow, verbose);
    m_bndGsh = Gs::Init(tmp2, vCommRow, verbose);
    Gs::Unique(tmp, vCommRow);
    for (unsigned int i = 0; i < m_numGlobalCoeffs; ++i)
    {
        m_globalToUniversalMapUnique[i] = (tmp[i] >= 0 ? 1 : 0);
    }
    for (unsigned int i = 0; i < m_numGlobalBndCoeffs; ++i)
    {
        m_globalToUniversalBndMapUnique[i] = (tmp2[i] >= 0 ? 1 : 0);
    }
}
 
/**
 * @brief Construct an AssemblyMapCG object which corresponds to the
 * linear space of the current object.
 *
 * This function is used to create a linear-space assembly map, which is
 * then used in the linear space preconditioner in the conjugate
 * gradient solve.
 */
AssemblyMapSharedPtr AssemblyMapCG::v_LinearSpaceMap(const ExpList &locexp,
                                                     GlobalSysSolnType solnType)
{
    AssemblyMapCGSharedPtr returnval;
 
    int i, j;
    int nverts                                               = 0;
    const std::shared_ptr<LocalRegions::ExpansionVector> exp = locexp.GetExp();
    int nelmts                                               = exp->size();
    const bool verbose = locexp.GetSession()->DefinesCmdLineArgument("verbose");
 
    // Get Default Map and turn off any searched values.
    returnval = MemoryManager<AssemblyMapCG>::AllocateSharedPtr(
        m_session, locexp.GetComm());
    returnval->m_solnType           = solnType;
    returnval->m_preconType         = "Null";
    returnval->m_maxStaticCondLevel = 0;
    returnval->m_signChange         = false;
    returnval->m_comm               = m_comm;
 
    // Count the number of vertices
    for (i = 0; i < nelmts; ++i)
    {
        nverts += (*exp)[i]->GetNverts();
    }
 
    returnval->m_numLocalCoeffs   = nverts;
    returnval->m_localToGlobalMap = Array<OneD, int>(nverts, -1);
 
    // Store original global ids in this map
    returnval->m_localToGlobalBndMap = Array<OneD, int>(nverts, -1);
 
    int cnt  = 0;
    int cnt1 = 0;
    Array<OneD, int> GlobCoeffs(m_numGlobalCoeffs, -1);
 
    // Set up local to global map;
    for (i = 0; i < nelmts; ++i)
    {
        for (j = 0; j < (*exp)[i]->GetNverts(); ++j)
        {
            returnval->m_localToGlobalMap[cnt] =
                returnval->m_localToGlobalBndMap[cnt] =
                    m_localToGlobalMap[cnt1 + (*exp)[i]->GetVertexMap(j, true)];
            GlobCoeffs[returnval->m_localToGlobalMap[cnt]] = 1;
 
            // Set up numLocalDirBndCoeffs
            if ((returnval->m_localToGlobalMap[cnt]) < m_numGlobalDirBndCoeffs)
            {
                returnval->m_numLocalDirBndCoeffs++;
            }
            cnt++;
        }
        cnt1 += (*exp)[i]->GetNcoeffs();
    }
 
    cnt = 0;
    // Reset global numbering and count number of dofs
    for (i = 0; i < m_numGlobalCoeffs; ++i)
    {
        if (GlobCoeffs[i] != -1)
        {
            GlobCoeffs[i] = cnt++;
        }
    }
 
    // Set up number of globalCoeffs;
    returnval->m_numGlobalCoeffs = cnt;
 
    // Set up number of global Dirichlet boundary coefficients
    for (i = 0; i < m_numGlobalDirBndCoeffs; ++i)
    {
        if (GlobCoeffs[i] != -1)
        {
            returnval->m_numGlobalDirBndCoeffs++;
        }
    }
 
    // Set up global to universal map
    if (m_globalToUniversalMap.size())
    {
        LibUtilities::CommSharedPtr vCommRow =
            m_session->GetComm()->GetRowComm();
        int nglocoeffs                          = returnval->m_numGlobalCoeffs;
        returnval->m_globalToUniversalMap       = Array<OneD, int>(nglocoeffs);
        returnval->m_globalToUniversalMapUnique = Array<OneD, int>(nglocoeffs);
 
        // Reset local to global map and setup universal map
        for (i = 0; i < nverts; ++i)
        {
            cnt                              = returnval->m_localToGlobalMap[i];
            returnval->m_localToGlobalMap[i] = GlobCoeffs[cnt];
 
            returnval->m_globalToUniversalMap[GlobCoeffs[cnt]] =
                m_globalToUniversalMap[cnt];
        }
 
        Nektar::Array<OneD, long> tmp(nglocoeffs);
        Vmath::Zero(nglocoeffs, tmp, 1);
        for (unsigned int i = 0; i < nglocoeffs; ++i)
        {
            tmp[i] = returnval->m_globalToUniversalMap[i];
        }
        returnval->m_gsh = Gs::Init(tmp, vCommRow, verbose);
        Gs::Unique(tmp, vCommRow);
        for (unsigned int i = 0; i < nglocoeffs; ++i)
        {
            returnval->m_globalToUniversalMapUnique[i] = (tmp[i] >= 0 ? 1 : 0);
        }
    }
    else // not sure this option is ever needed.
    {
        for (i = 0; i < nverts; ++i)
        {
            cnt                              = returnval->m_localToGlobalMap[i];
            returnval->m_localToGlobalMap[i] = GlobCoeffs[cnt];
        }
    }
 
    return returnval;
}
 
/**
 * The bandwidth calculated here corresponds to what is referred to as
 * half-bandwidth.  If the elements of the matrix are designated as
 * a_ij, it corresponds to the maximum value of |i-j| for non-zero
 * a_ij.  As a result, the value also corresponds to the number of
 * sub- or super-diagonals.
 *
 * The bandwith can be calculated elementally as it corresponds to the
 * maximal elemental bandwith (i.e. the maximal difference in global
 * DOF index for every element).
 *
 * We caluclate here the bandwith of the full global system.
 */
void AssemblyMapCG::CalculateFullSystemBandWidth()
{
    int i, j;
    int cnt = 0;
    int locSize;
    int maxId;
    int minId;
    int bwidth = -1;
    for (i = 0; i < m_numPatches; ++i)
    {
        locSize =
            m_numLocalBndCoeffsPerPatch[i] + m_numLocalIntCoeffsPerPatch[i];
        maxId = -1;
        minId = m_numLocalCoeffs + 1;
        for (j = 0; j < locSize; j++)
        {
            if (m_localToGlobalMap[cnt + j] >= m_numGlobalDirBndCoeffs)
            {
                if (m_localToGlobalMap[cnt + j] > maxId)
                {
                    maxId = m_localToGlobalMap[cnt + j];
                }
 
                if (m_localToGlobalMap[cnt + j] < minId)
                {
                    minId = m_localToGlobalMap[cnt + j];
                }
            }
        }
        bwidth = (bwidth > (maxId - minId)) ? bwidth : (maxId - minId);
 
        cnt += locSize;
    }
 
    m_fullSystemBandWidth = bwidth;
}
 
int AssemblyMapCG::v_GetLocalToGlobalMap(const int i) const
{
    return m_localToGlobalMap[i];
}
 
int AssemblyMapCG::v_GetGlobalToUniversalMap(const int i) const
{
    return m_globalToUniversalMap[i];
}
 
int AssemblyMapCG::v_GetGlobalToUniversalMapUnique(const int i) const
{
    return m_globalToUniversalMapUnique[i];
}
 
const Array<OneD, const int> &AssemblyMapCG::v_GetLocalToGlobalMap(void)
{
    return m_localToGlobalMap;
}
 
const Array<OneD, const int> &AssemblyMapCG::v_GetGlobalToUniversalMap(void)
{
    return m_globalToUniversalMap;
}
 
const Array<OneD, const int> &AssemblyMapCG::v_GetGlobalToUniversalMapUnique(
    void)
{
    return m_globalToUniversalMapUnique;
}
 
NekDouble AssemblyMapCG::v_GetLocalToGlobalSign(const int i) const
{
    if (m_signChange)
    {
        return m_localToGlobalSign[i];
    }
    else
    {
        return 1.0;
    }
}
 
const Array<OneD, NekDouble> &AssemblyMapCG::v_GetLocalToGlobalSign() const
{
    return m_localToGlobalSign;
}
 
void AssemblyMapCG::v_LocalToGlobal(const Array<OneD, const NekDouble> &loc,
                                    Array<OneD, NekDouble> &global,
                                    bool useComm) const
{
    Array<OneD, const NekDouble> local;
    if (global.data() == loc.data())
    {
        local = Array<OneD, NekDouble>(m_numLocalCoeffs, loc.data());
    }
    else
    {
        local = loc; // create reference
    }
 
    if (m_signChange)
    {
        Vmath::Scatr(m_numLocalCoeffs, m_localToGlobalSign.get(), local.get(),
                     m_localToGlobalMap.get(), global.get());
    }
    else
    {
        Vmath::Scatr(m_numLocalCoeffs, local.get(), m_localToGlobalMap.get(),
                     global.get());
    }
 
    // ensure all values are unique by calling a max
    if (useComm)
    {
        Gs::Gather(global, Gs::gs_max, m_gsh);
    }
}
 
void AssemblyMapCG::v_LocalToGlobal(const NekVector<NekDouble> &loc,
                                    NekVector<NekDouble> &global,
                                    bool useComm) const
{
    LocalToGlobal(loc.GetPtr(), global.GetPtr(), useComm);
}
 
void AssemblyMapCG::v_GlobalToLocal(const Array<OneD, const NekDouble> &global,
                                    Array<OneD, NekDouble> &loc) const
{
    Array<OneD, const NekDouble> glo;
    if (global.data() == loc.data())
    {
        glo = Array<OneD, NekDouble>(m_numGlobalCoeffs, global.data());
    }
    else
    {
        glo = global; // create reference
    }
 
    if (m_signChange)
    {
        Vmath::Gathr(m_numLocalCoeffs, m_localToGlobalSign.get(), glo.get(),
                     m_localToGlobalMap.get(), loc.get());
    }
    else
    {
        Vmath::Gathr(m_numLocalCoeffs, glo.get(), m_localToGlobalMap.get(),
                     loc.get());
    }
}
 
void AssemblyMapCG::v_GlobalToLocal(const NekVector<NekDouble> &global,
                                    NekVector<NekDouble> &loc) const
{
    GlobalToLocal(global.GetPtr(), loc.GetPtr());
}
 
void AssemblyMapCG::v_Assemble(const Array<OneD, const NekDouble> &loc,
                               Array<OneD, NekDouble> &global) const
{
    Array<OneD, const NekDouble> local;
    if (global.data() == loc.data())
    {
        local = Array<OneD, NekDouble>(m_numLocalCoeffs, loc.data());
    }
    else
    {
        local = loc; // create reference
    }
 
    Vmath::Zero(m_numGlobalCoeffs, global.get(), 1);
 
    if (m_signChange)
    {
        Vmath::Assmb(m_numLocalCoeffs, m_localToGlobalSign.get(), local.get(),
                     m_localToGlobalMap.get(), global.get());
    }
    else
    {
        Vmath::Assmb(m_numLocalCoeffs, local.get(), m_localToGlobalMap.get(),
                     global.get());
    }
    UniversalAssemble(global);
}
 
void AssemblyMapCG::v_Assemble(const NekVector<NekDouble> &loc,
                               NekVector<NekDouble> &global) const
{
    Assemble(loc.GetPtr(), global.GetPtr());
}
 
void AssemblyMapCG::v_UniversalAssemble(Array<OneD, NekDouble> &pGlobal) const
{
    Gs::Gather(pGlobal, Gs::gs_add, m_gsh);
}
 
void AssemblyMapCG::v_UniversalAssemble(NekVector<NekDouble> &pGlobal) const
{
    UniversalAssemble(pGlobal.GetPtr());
}
 
void AssemblyMapCG::v_UniversalAssemble(Array<OneD, NekDouble> &pGlobal,
                                        int offset) const
{
    Array<OneD, NekDouble> tmp(offset);
    Vmath::Vcopy(offset, pGlobal, 1, tmp, 1);
    UniversalAssemble(pGlobal);
    Vmath::Vcopy(offset, tmp, 1, pGlobal, 1);
}
 
int AssemblyMapCG::v_GetFullSystemBandWidth() const
{
    return m_fullSystemBandWidth;
}
 
int AssemblyMapCG::v_GetNumNonDirVertexModes() const
{
    return m_numNonDirVertexModes;
}
 
int AssemblyMapCG::v_GetNumNonDirEdgeModes() const
{
    return m_numNonDirEdgeModes;
}
 
int AssemblyMapCG::v_GetNumNonDirFaceModes() const
{
    return m_numNonDirFaceModes;
}
 
int AssemblyMapCG::v_GetNumDirEdges() const
{
    return m_numDirEdges;
}
 
int AssemblyMapCG::v_GetNumDirFaces() const
{
    return m_numDirFaces;
}
 
int AssemblyMapCG::v_GetNumNonDirEdges() const
{
    return m_numNonDirEdges;
}
 
int AssemblyMapCG::v_GetNumNonDirFaces() const
{
    return m_numNonDirFaces;
}
 
const Array<OneD, const int> &AssemblyMapCG::v_GetExtraDirEdges()
{
    return m_extraDirEdges;
}
} // namespace Nektar::MultiRegions