Intrepid2_IntegratedLegendreBasis_HGRAD_TET.hpp

Go to the documentation of this file.

// @HEADER
// ************************************************************************
//
//                           Intrepid2 Package
//                 Copyright (2007) Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
// license for use of this work by or on behalf of the U.S. Government.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// 3. Neither the name of the Corporation nor the names of the
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
// PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
// LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Questions? Contact Kyungjoo Kim  (kyukim@sandia.gov),
//                    Mauro Perego  (mperego@sandia.gov), or
//                    Nate Roberts  (nvrober@sandia.gov)
//
// ************************************************************************
// @HEADER

#ifndef Intrepid2_IntegratedLegendreBasis_HGRAD_TET_h
#define Intrepid2_IntegratedLegendreBasis_HGRAD_TET_h

#include <Kokkos_View.hpp>
#include <Kokkos_DynRankView.hpp>

#include <Intrepid2_config.h>

#include "Intrepid2_Polynomials.hpp"
#include "Intrepid2_Utils.hpp"

namespace Intrepid2
{
  template<class ExecutionSpace, class OutputScalar, class PointScalar,
           class OutputFieldType, class InputPointsType>
  struct Hierarchical_HGRAD_TET_Functor
  {
    using ScratchSpace        = typename ExecutionSpace::scratch_memory_space;
    using OutputScratchView   = Kokkos::View<OutputScalar*,ScratchSpace,Kokkos::MemoryTraits<Kokkos::Unmanaged>>;
    using OutputScratchView2D = Kokkos::View<OutputScalar**,ScratchSpace,Kokkos::MemoryTraits<Kokkos::Unmanaged>>;
    using PointScratchView    = Kokkos::View<PointScalar*, ScratchSpace,Kokkos::MemoryTraits<Kokkos::Unmanaged>>;
    
    using TeamPolicy = Kokkos::TeamPolicy<ExecutionSpace>;
    using TeamMember = typename TeamPolicy::member_type;
    
    EOperator opType_;
    
    OutputFieldType  output_;      // F,P
    InputPointsType  inputPoints_; // P,D
    
    int polyOrder_;
    bool defineVertexFunctions_;
    int numFields_, numPoints_;
    
    size_t fad_size_output_;
    
    static const int numVertices = 4;
    static const int numEdges    = 6;
    // the following ordering of the edges matches that used by ESEAS
    const int edge_start_[numEdges] = {0,1,0,0,1,2}; // edge i is from edge_start_[i] to edge_end_[i]
    const int edge_end_[numEdges]   = {1,2,2,3,3,3}; // edge i is from edge_start_[i] to edge_end_[i]
    
    static const int numFaces    = 4;
    const int face_vertex_0[numFaces] = {0,0,1,0}; // faces are abc where 0 ≤ a < b < c ≤ 3
    const int face_vertex_1[numFaces] = {1,1,2,2};
    const int face_vertex_2[numFaces] = {2,3,3,3};
    
    // this allows us to look up the edge ordinal of the first edge of a face
    // this is useful because face functions are defined using edge basis functions of the first edge of the face
    const int face_ordinal_of_first_edge[numFaces] = {0,0,1,2};
    
    Hierarchical_HGRAD_TET_Functor(EOperator opType, OutputFieldType output, InputPointsType inputPoints,
                                    int polyOrder, bool defineVertexFunctions)
    : opType_(opType), output_(output), inputPoints_(inputPoints),
      polyOrder_(polyOrder), defineVertexFunctions_(defineVertexFunctions),
      fad_size_output_(getScalarDimensionForView(output))
    {
      numFields_ = output.extent_int(0);
      numPoints_ = output.extent_int(1);
      INTREPID2_TEST_FOR_EXCEPTION(numPoints_ != inputPoints.extent_int(0), std::invalid_argument, "point counts need to match!");
      INTREPID2_TEST_FOR_EXCEPTION(numFields_ != (polyOrder_+1)*(polyOrder_+2)*(polyOrder_+3)/6, std::invalid_argument, "output field size does not match basis cardinality");
    }
    
    KOKKOS_INLINE_FUNCTION
    void operator()( const TeamMember & teamMember ) const
    {
      const int numFaceBasisFunctionsPerFace = (polyOrder_-2) * (polyOrder_-1) / 2;
      const int numInteriorBasisFunctions = (polyOrder_-3) * (polyOrder_-2) * (polyOrder_-1) / 6;
      
      auto pointOrdinal = teamMember.league_rank();
      OutputScratchView legendre_values1_at_point, legendre_values2_at_point;
      OutputScratchView2D jacobi_values1_at_point, jacobi_values2_at_point, jacobi_values3_at_point;
      const int numAlphaValues = (polyOrder_-1 > 1) ? (polyOrder_-1) : 1; // make numAlphaValues at least 1 so we can avoid zero-extent allocations…
      if (fad_size_output_ > 0) {
        legendre_values1_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
        legendre_values2_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
        jacobi_values1_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1, fad_size_output_);
        jacobi_values2_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1, fad_size_output_);
        jacobi_values3_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1, fad_size_output_);
      }
      else {
        legendre_values1_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
        legendre_values2_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
        jacobi_values1_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1);
        jacobi_values2_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1);
        jacobi_values3_at_point   = OutputScratchView2D(teamMember.team_shmem(), numAlphaValues, polyOrder_ + 1);
      }
      
      const auto & x = inputPoints_(pointOrdinal,0);
      const auto & y = inputPoints_(pointOrdinal,1);
      const auto & z = inputPoints_(pointOrdinal,2);
      
      // write as barycentric coordinates:
      const PointScalar lambda[numVertices] = {1. - x - y - z, x, y, z};
      const PointScalar lambda_dx[numVertices] = {-1., 1., 0., 0.};
      const PointScalar lambda_dy[numVertices] = {-1., 0., 1., 0.};
      const PointScalar lambda_dz[numVertices] = {-1., 0., 0., 1.};
      
      const int num1DEdgeFunctions = polyOrder_ - 1;
      
      switch (opType_)
      {
        case OPERATOR_VALUE:
        {
          // vertex functions come first, according to vertex ordering: (0,0,0), (1,0,0), (0,1,0), (0,0,1)
          for (int vertexOrdinal=0; vertexOrdinal<numVertices; vertexOrdinal++)
          {
            output_(vertexOrdinal,pointOrdinal) = lambda[vertexOrdinal];
          }
          if (!defineVertexFunctions_)
          {
            // "DG" basis case
            // here, we overwrite the first vertex function with 1:
            output_(0,pointOrdinal) = 1.0;
          }
          
          // edge functions
          int fieldOrdinalOffset = numVertices;
          for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
          {
            const auto & s0 = lambda[edge_start_[edgeOrdinal]];
            const auto & s1 = lambda[  edge_end_[edgeOrdinal]];

            Polynomials::shiftedScaledIntegratedLegendreValues(legendre_values1_at_point, polyOrder_, PointScalar(s1), PointScalar(s0+s1));
            for (int edgeFunctionOrdinal=0; edgeFunctionOrdinal<num1DEdgeFunctions; edgeFunctionOrdinal++)
            {
              // the first two integrated legendre functions are essentially the vertex functions; hence the +2 on on the RHS here:
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal) = legendre_values1_at_point(edgeFunctionOrdinal+2);
            }
            fieldOrdinalOffset += num1DEdgeFunctions;
          }
          /*
           Face functions for face abc are the product of edge functions on their ab edge
           and a Jacobi polynomial [L^2i_j](s0+s1,s2) = L^2i_j(s2;s0+s1+s2)
           */
          for (int faceOrdinal=0; faceOrdinal<numFaces; faceOrdinal++)
          {
            const auto & s0 = lambda[face_vertex_0[faceOrdinal]];
            const auto & s1 = lambda[face_vertex_1[faceOrdinal]];
            const auto & s2 = lambda[face_vertex_2[faceOrdinal]];
            const PointScalar jacobiScaling = s0 + s1 + s2;
            
            // compute integrated Jacobi values for each desired value of alpha
            for (int n=2; n<=polyOrder_; n++)
            {
              const double alpha = n*2;
              const int alphaOrdinal = n-2;
              using Kokkos::subview;
              using Kokkos::ALL;
              auto jacobi_alpha = subview(jacobi_values1_at_point, alphaOrdinal, ALL);
              Polynomials::integratedJacobiValues(jacobi_alpha, alpha, polyOrder_-2, s2, jacobiScaling);
            }
            
            const int edgeOrdinal = face_ordinal_of_first_edge[faceOrdinal];
            int localFaceBasisOrdinal = 0;
            for (int totalPolyOrder=3; totalPolyOrder<=polyOrder_; totalPolyOrder++)
            {
              for (int i=2; i<totalPolyOrder; i++)
              {
                const int edgeBasisOrdinal = edgeOrdinal*num1DEdgeFunctions + i-2 + numVertices;
                const auto & edgeValue = output_(edgeBasisOrdinal,pointOrdinal);
                const int alphaOrdinal = i-2;
                
                const int j = totalPolyOrder - i;
                const auto & jacobiValue = jacobi_values1_at_point(alphaOrdinal,j);
                const int fieldOrdinal = fieldOrdinalOffset + localFaceBasisOrdinal;
                output_(fieldOrdinal,pointOrdinal) = edgeValue * jacobiValue;
                
                localFaceBasisOrdinal++;
              }
            }
            fieldOrdinalOffset += numFaceBasisFunctionsPerFace;
          }
          // interior functions
          // compute integrated Jacobi values for each desired value of alpha
          for (int n=3; n<=polyOrder_; n++)
          {
            const double alpha = n*2;
            const double jacobiScaling = 1.0;
            const int alphaOrdinal = n-3;
            using Kokkos::subview;
            using Kokkos::ALL;
            auto jacobi_alpha = subview(jacobi_values1_at_point, alphaOrdinal, ALL);
            Polynomials::integratedJacobiValues(jacobi_alpha, alpha, polyOrder_-3, lambda[3], jacobiScaling);
          }
          const int min_i  = 2;
          const int min_j  = 1;
          const int min_k  = 1;
          const int min_ij = min_i + min_j;
          const int min_ijk = min_ij + min_k;
          int localInteriorBasisOrdinal = 0;
          for (int totalPolyOrder_ijk=min_ijk; totalPolyOrder_ijk <= polyOrder_; totalPolyOrder_ijk++)
          {
            int localFaceBasisOrdinal = 0;
            for (int totalPolyOrder_ij=min_ij; totalPolyOrder_ij <= totalPolyOrder_ijk-min_j; totalPolyOrder_ij++)
            {
              for (int i=2; i <= totalPolyOrder_ij-min_j; i++)
              {
                const int j = totalPolyOrder_ij - i;
                const int k = totalPolyOrder_ijk - totalPolyOrder_ij;
                const int faceBasisOrdinal = numEdges*num1DEdgeFunctions + numVertices + localFaceBasisOrdinal;
                const auto & faceValue = output_(faceBasisOrdinal,pointOrdinal);
                const int alphaOrdinal = (i+j)-3;
                localFaceBasisOrdinal++;
              
                const int fieldOrdinal = fieldOrdinalOffset + localInteriorBasisOrdinal;
                const auto & jacobiValue = jacobi_values1_at_point(alphaOrdinal,k);
                output_(fieldOrdinal,pointOrdinal) = faceValue * jacobiValue;
                localInteriorBasisOrdinal++;
              } // end i loop
            } // end totalPolyOrder_ij loop
          } // end totalPolyOrder_ijk loop
          fieldOrdinalOffset += numInteriorBasisFunctions;
        } // end OPERATOR_VALUE
          break;
        case OPERATOR_GRAD:
        case OPERATOR_D1:
        {
          // vertex functions
          if (defineVertexFunctions_)
          {
            // standard, "CG" basis case
            // first vertex function is 1-x-y-z
            output_(0,pointOrdinal,0) = -1.0;
            output_(0,pointOrdinal,1) = -1.0;
            output_(0,pointOrdinal,2) = -1.0;
          }
          else
          {
            // "DG" basis case
            // here, the first "vertex" function is 1, so the derivative is 0:
            output_(0,pointOrdinal,0) = 0.0;
            output_(0,pointOrdinal,1) = 0.0;
            output_(0,pointOrdinal,2) = 0.0;
          }
          // second vertex function is x
          output_(1,pointOrdinal,0) = 1.0;
          output_(1,pointOrdinal,1) = 0.0;
          output_(1,pointOrdinal,2) = 0.0;
          // third vertex function is y
          output_(2,pointOrdinal,0) = 0.0;
          output_(2,pointOrdinal,1) = 1.0;
          output_(2,pointOrdinal,2) = 0.0;
          // fourth vertex function is z
          output_(3,pointOrdinal,0) = 0.0;
          output_(3,pointOrdinal,1) = 0.0;
          output_(3,pointOrdinal,2) = 1.0;

          // edge functions
          int fieldOrdinalOffset = numVertices;
          /*
           Per Fuentes et al. (see Appendix E.1, E.2), the edge functions, defined for i ≥ 2, are
             [L_i](s0,s1) = L_i(s1; s0+s1)
           and have gradients:
             grad [L_i](s0,s1) = [P_{i-1}](s0,s1) grad s1 + [R_{i-1}](s0,s1) grad (s0 + s1)
           where
             [R_{i-1}](s0,s1) = R_{i-1}(s1; s0+s1) = d/dt L_{i}(s0; s0+s1)
           The P_i we have implemented in shiftedScaledLegendreValues, while d/dt L_{i+1} is
           implemented in shiftedScaledIntegratedLegendreValues_dt.
           */
          // rename the scratch memory to match our usage here:
          auto & P_i_minus_1 = legendre_values1_at_point;
          auto & L_i_dt      = legendre_values2_at_point;
          for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
          {
            const auto & s0 = lambda[edge_start_[edgeOrdinal]];
            const auto & s1 = lambda[  edge_end_[edgeOrdinal]];
            
            const auto & s0_dx = lambda_dx[edge_start_[edgeOrdinal]];
            const auto & s0_dy = lambda_dy[edge_start_[edgeOrdinal]];
            const auto & s0_dz = lambda_dz[edge_start_[edgeOrdinal]];
            const auto & s1_dx = lambda_dx[  edge_end_[edgeOrdinal]];
            const auto & s1_dy = lambda_dy[  edge_end_[edgeOrdinal]];
            const auto & s1_dz = lambda_dz[  edge_end_[edgeOrdinal]];
            
            Polynomials::shiftedScaledLegendreValues             (P_i_minus_1, polyOrder_-1, PointScalar(s1), PointScalar(s0+s1));
            Polynomials::shiftedScaledIntegratedLegendreValues_dt(L_i_dt,      polyOrder_,   PointScalar(s1), PointScalar(s0+s1));
            for (int edgeFunctionOrdinal=0; edgeFunctionOrdinal<num1DEdgeFunctions; edgeFunctionOrdinal++)
            {
              // the first two (integrated) Legendre functions are essentially the vertex functions; hence the +2 here:
              const int i = edgeFunctionOrdinal+2;
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal,0) = P_i_minus_1(i-1) * s1_dx + L_i_dt(i) * (s1_dx + s0_dx);
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal,1) = P_i_minus_1(i-1) * s1_dy + L_i_dt(i) * (s1_dy + s0_dy);
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal,2) = P_i_minus_1(i-1) * s1_dz + L_i_dt(i) * (s1_dz + s0_dz);
            }
            fieldOrdinalOffset += num1DEdgeFunctions;
          }
          
          /*
           Fuentes et al give the face functions as phi_{ij}, with gradient:
             grad phi_{ij}(s0,s1,s2) = [L^{2i}_j](s0+s1,s2) grad [L_i](s0,s1) + [L_i](s0,s1) grad [L^{2i}_j](s0+s1,s2)
           where:
           - grad [L_i](s0,s1) is the edge function gradient we computed above
           - [L_i](s0,s1) is the edge function which we have implemented above (in OPERATOR_VALUE)
           - L^{2i}_j is a Jacobi polynomial with:
               [L^{2i}_j](s0,s1) = L^{2i}_j(s1;s0+s1)
             and the gradient for j ≥ 1 is
               grad [L^{2i}_j](s0,s1) = [P^{2i}_{j-1}](s0,s1) grad s1 + [R^{2i}_{j-1}(s0,s1)] grad (s0 + s1)
           Here,
             [P^{2i}_{j-1}](s0,s1) = P^{2i}_{j-1}(s1,s0+s1)
           and
             [R^{2i}_{j-1}(s0,s1)] = d/dt L^{2i}_j(s1,s0+s1)
           We have implemented P^{alpha}_{j} as shiftedScaledJacobiValues,
           and d/dt L^{alpha}_{j} as integratedJacobiValues_dt.
           */
          // rename the scratch memory to match our usage here:
          auto & L_i            = legendre_values2_at_point;
          auto & L_2i_j_dt      = jacobi_values1_at_point;
          auto & L_2i_j         = jacobi_values2_at_point;
          auto & P_2i_j_minus_1 = jacobi_values3_at_point;
          
          for (int faceOrdinal=0; faceOrdinal<numFaces; faceOrdinal++)
          {
            const auto & s0 = lambda[face_vertex_0[faceOrdinal]];
            const auto & s1 = lambda[face_vertex_1[faceOrdinal]];
            const auto & s2 = lambda[face_vertex_2[faceOrdinal]];
            Polynomials::shiftedScaledIntegratedLegendreValues(L_i, polyOrder_, s1, s0+s1);
            
            const PointScalar jacobiScaling = s0 + s1 + s2;
            
            // compute integrated Jacobi values for each desired value of alpha
            for (int n=2; n<=polyOrder_; n++)
            {
              const double alpha = n*2;
              const int alphaOrdinal = n-2;
              using Kokkos::subview;
              using Kokkos::ALL;
              auto L_2i_j_dt_alpha      = subview(L_2i_j_dt,      alphaOrdinal, ALL);
              auto L_2i_j_alpha         = subview(L_2i_j,         alphaOrdinal, ALL);
              auto P_2i_j_minus_1_alpha = subview(P_2i_j_minus_1, alphaOrdinal, ALL);
              Polynomials::integratedJacobiValues_dt(L_2i_j_dt_alpha,      alpha, polyOrder_-2, s2, jacobiScaling);
              Polynomials::integratedJacobiValues   (L_2i_j_alpha,         alpha, polyOrder_-2, s2, jacobiScaling);
              Polynomials::shiftedScaledJacobiValues(P_2i_j_minus_1_alpha, alpha, polyOrder_-1, s2, jacobiScaling);
            }
            
            const int edgeOrdinal = face_ordinal_of_first_edge[faceOrdinal];
            int localFaceOrdinal = 0;
            for (int totalPolyOrder=3; totalPolyOrder<=polyOrder_; totalPolyOrder++)
            {
              for (int i=2; i<totalPolyOrder; i++)
              {
                const int edgeBasisOrdinal = edgeOrdinal*num1DEdgeFunctions + i-2 + numVertices;
                const auto & grad_L_i_dx = output_(edgeBasisOrdinal,pointOrdinal,0);
                const auto & grad_L_i_dy = output_(edgeBasisOrdinal,pointOrdinal,1);
                const auto & grad_L_i_dz = output_(edgeBasisOrdinal,pointOrdinal,2);
                
                const int alphaOrdinal = i-2;
                
                const auto & s0_dx = lambda_dx[face_vertex_0[faceOrdinal]];
                const auto & s0_dy = lambda_dy[face_vertex_0[faceOrdinal]];
                const auto & s0_dz = lambda_dz[face_vertex_0[faceOrdinal]];
                const auto & s1_dx = lambda_dx[face_vertex_1[faceOrdinal]];
                const auto & s1_dy = lambda_dy[face_vertex_1[faceOrdinal]];
                const auto & s1_dz = lambda_dz[face_vertex_1[faceOrdinal]];
                const auto & s2_dx = lambda_dx[face_vertex_2[faceOrdinal]];
                const auto & s2_dy = lambda_dy[face_vertex_2[faceOrdinal]];
                const auto & s2_dz = lambda_dz[face_vertex_2[faceOrdinal]];
                
                int j = totalPolyOrder - i;
 
                // put references to the entries of interest in like-named variables with lowercase first letters
                auto & l_2i_j         = L_2i_j(alphaOrdinal,j);
                auto & l_i            = L_i(i);
                auto & l_2i_j_dt      = L_2i_j_dt(alphaOrdinal,j);
                auto & p_2i_j_minus_1 = P_2i_j_minus_1(alphaOrdinal,j-1);
                
                const OutputScalar basisValue_dx = l_2i_j * grad_L_i_dx + l_i * (p_2i_j_minus_1 * s2_dx + l_2i_j_dt * (s0_dx + s1_dx + s2_dx));
                const OutputScalar basisValue_dy = l_2i_j * grad_L_i_dy + l_i * (p_2i_j_minus_1 * s2_dy + l_2i_j_dt * (s0_dy + s1_dy + s2_dy));
                const OutputScalar basisValue_dz = l_2i_j * grad_L_i_dz + l_i * (p_2i_j_minus_1 * s2_dz + l_2i_j_dt * (s0_dz + s1_dz + s2_dz));
                
                const int fieldOrdinal = fieldOrdinalOffset + localFaceOrdinal;
                
                output_(fieldOrdinal,pointOrdinal,0) = basisValue_dx;
                output_(fieldOrdinal,pointOrdinal,1) = basisValue_dy;
                output_(fieldOrdinal,pointOrdinal,2) = basisValue_dz;
                
                localFaceOrdinal++;
              }
            }
            fieldOrdinalOffset += numFaceBasisFunctionsPerFace;
          }
          // interior functions
          /*
           Per Fuentes et al. (see Appendix E.1, E.2), the interior functions, defined for i ≥ 2, are
             phi_ij(lambda_012) [L^{2(i+j)}_k](1-lambda_3,lambda_3) = phi_ij(lambda_012) L^{2(i+j)}_k (lambda_3; 1)
           and have gradients:
             L^{2(i+j)}_k (lambda_3; 1) grad (phi_ij(lambda_012)) + phi_ij(lambda_012) grad (L^{2(i+j)}_k (lambda_3; 1))
           where:
             - phi_ij(lambda_012) is the (i,j) basis function on face 012,
             - L^{alpha}_j(t0; t1) is the jth integrated Jacobi polynomial
           and the gradient of the integrated Jacobi polynomial can be computed:
           - grad L^{alpha}_j(t0; t1) = P^{alpha}_{j-1} (t0;t1) grad t0 + R^{alpha}_{j-1}(t0,t1) grad t1
           Here, t1=1, so this simplifies to:
           - grad L^{alpha}_j(t0; t1) = P^{alpha}_{j-1} (t0;t1) grad t0
           
           The P_i we have implemented in shiftedScaledJacobiValues.
           */
          // rename the scratch memory to match our usage here:
          auto & L_alpha = jacobi_values1_at_point;
          auto & P_alpha = jacobi_values2_at_point;
          
          // precompute values used in face ordinal 0:
          {
            const auto & s0 = lambda[0];
            const auto & s1 = lambda[1];
            const auto & s2 = lambda[2];
            // Legendre:
            Polynomials::shiftedScaledIntegratedLegendreValues(legendre_values1_at_point, polyOrder_, PointScalar(s1), PointScalar(s0+s1));
            
            // Jacobi for each desired alpha value:
            const PointScalar jacobiScaling = s0 + s1 + s2;
            for (int n=2; n<=polyOrder_; n++)
            {
              const double alpha = n*2;
              const int alphaOrdinal = n-2;
              using Kokkos::subview;
              using Kokkos::ALL;
              auto jacobi_alpha = subview(jacobi_values3_at_point, alphaOrdinal, ALL);
              Polynomials::integratedJacobiValues(jacobi_alpha, alpha, polyOrder_-2, s2, jacobiScaling);
            }
          }
          
          // interior
          for (int n=3; n<=polyOrder_; n++)
          {
            const double alpha = n*2;
            const double jacobiScaling = 1.0;
            const int alphaOrdinal = n-3;
            using Kokkos::subview;
            using Kokkos::ALL;
            
            // values for interior functions:
            auto L = subview(L_alpha, alphaOrdinal, ALL);
            auto P = subview(P_alpha, alphaOrdinal, ALL);
            Polynomials::integratedJacobiValues   (L, alpha, polyOrder_-3, lambda[3], jacobiScaling);
            Polynomials::shiftedScaledJacobiValues(P, alpha, polyOrder_-3, lambda[3], jacobiScaling);
          }
          
          const int min_i  = 2;
          const int min_j  = 1;
          const int min_k  = 1;
          const int min_ij = min_i + min_j;
          const int min_ijk = min_ij + min_k;
          int localInteriorBasisOrdinal = 0;
          for (int totalPolyOrder_ijk=min_ijk; totalPolyOrder_ijk <= polyOrder_; totalPolyOrder_ijk++)
          {
            int localFaceBasisOrdinal = 0;
            for (int totalPolyOrder_ij=min_ij; totalPolyOrder_ij <= totalPolyOrder_ijk-min_j; totalPolyOrder_ij++)
            {
              for (int i=2; i <= totalPolyOrder_ij-min_j; i++)
              {
                const int j = totalPolyOrder_ij - i;
                const int k = totalPolyOrder_ijk - totalPolyOrder_ij;
                // interior functions use basis values belonging to the first face, 012
                const int faceBasisOrdinal = numEdges*num1DEdgeFunctions + numVertices + localFaceBasisOrdinal;
                
                const auto & faceValue_dx = output_(faceBasisOrdinal,pointOrdinal,0);
                const auto & faceValue_dy = output_(faceBasisOrdinal,pointOrdinal,1);
                const auto & faceValue_dz = output_(faceBasisOrdinal,pointOrdinal,2);
                
                // determine faceValue (on face 0)
                OutputScalar faceValue;
                {
                  const auto & edgeValue = legendre_values1_at_point(i);
                  const int alphaOrdinal = i-2;
                  const auto & jacobiValue = jacobi_values3_at_point(alphaOrdinal,j);
                  faceValue = edgeValue * jacobiValue;
                }
                localFaceBasisOrdinal++;
              
                const int alphaOrdinal = (i+j)-3;
              
                const int fieldOrdinal = fieldOrdinalOffset + localInteriorBasisOrdinal;
                const auto & integratedJacobiValue = L_alpha(alphaOrdinal,k);
                const auto & jacobiValue = P_alpha(alphaOrdinal,k-1);
                output_(fieldOrdinal,pointOrdinal,0) = integratedJacobiValue * faceValue_dx + faceValue * jacobiValue * lambda_dx[3];
                output_(fieldOrdinal,pointOrdinal,1) = integratedJacobiValue * faceValue_dy + faceValue * jacobiValue * lambda_dy[3];
                output_(fieldOrdinal,pointOrdinal,2) = integratedJacobiValue * faceValue_dz + faceValue * jacobiValue * lambda_dz[3];
                
                localInteriorBasisOrdinal++;
              } // end i loop
            } // end totalPolyOrder_ij loop
          } // end totalPolyOrder_ijk loop
          fieldOrdinalOffset += numInteriorBasisFunctions;
        }
          break;
        case OPERATOR_D2:
        case OPERATOR_D3:
        case OPERATOR_D4:
        case OPERATOR_D5:
        case OPERATOR_D6:
        case OPERATOR_D7:
        case OPERATOR_D8:
        case OPERATOR_D9:
        case OPERATOR_D10:
          INTREPID2_TEST_FOR_ABORT( true,
                                   ">>> ERROR: (Intrepid2::Basis_HGRAD_TET_Cn_FEM_ORTH::OrthPolynomialTri) Computing of second and higher-order derivatives is not currently supported");
        default:
          // unsupported operator type
          device_assert(false);
      }
    }
    
    // Provide the shared memory capacity.
    // This function takes the team_size as an argument,
    // which allows team_size-dependent allocations.
    size_t team_shmem_size (int team_size) const
    {
      // we will use shared memory to create a fast buffer for basis computations
      // for the (integrated) Legendre computations, we just need p+1 values stored
      // for the (integrated) Jacobi computations, though, we want (p+1)*(# alpha values)
      // alpha is either 2i or 2(i+j), where i=2,…,p or i+j=3,…,p.  So there are at most (p-1) alpha values needed.
      // We can have up to 3 of the (integrated) Jacobi values needed at once.
      const int numAlphaValues = std::max(polyOrder_-1, 1); // make it at least 1 so we can avoid zero-extent ranks…
      size_t shmem_size = 0;
      if (fad_size_output_ > 0)
      {
        // Legendre:
        shmem_size += 2 * OutputScratchView::shmem_size(polyOrder_ + 1, fad_size_output_);
        // Jacobi:
        shmem_size += 3 * OutputScratchView2D::shmem_size(numAlphaValues, polyOrder_ + 1, fad_size_output_);
      }
      else
      {
        // Legendre:
        shmem_size += 2 * OutputScratchView::shmem_size(polyOrder_ + 1);
        // Jacobi:
        shmem_size += 3 * OutputScratchView2D::shmem_size(numAlphaValues, polyOrder_ + 1);
      }
      
      return shmem_size;
    }
  };
  
  template<typename ExecutionSpace=Kokkos::DefaultExecutionSpace,
           typename OutputScalar = double,
           typename PointScalar  = double,
           bool defineVertexFunctions = true>  // if defineVertexFunctions is true, first four basis functions are 1-x-y-z, x, y, and z.  Otherwise, they are 1, x, y, and z.
  class IntegratedLegendreBasis_HGRAD_TET
  : public Basis<ExecutionSpace,OutputScalar,PointScalar>
  {
  public:
    using OrdinalTypeArray1DHost = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OrdinalTypeArray1DHost;
    using OrdinalTypeArray2DHost = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OrdinalTypeArray2DHost;
    
    using OutputViewType = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OutputViewType;
    using PointViewType  = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::PointViewType;
    using ScalarViewType = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::ScalarViewType;
  protected:
    int polyOrder_; // the maximum order of the polynomial
    bool defineVertexFunctions_;  // if true, first four basis functions are 1-x-y-z, x, y, and z.  Otherwise, they are 1, x, y, and z.
  public:
    IntegratedLegendreBasis_HGRAD_TET(int polyOrder)
    :
    polyOrder_(polyOrder)
    {
      this->basisCardinality_  = ((polyOrder+1) * (polyOrder+2) * (polyOrder+3)) / 6;
      this->basisDegree_       = polyOrder;
      this->basisCellTopology_ = shards::CellTopology(shards::getCellTopologyData<shards::Tetrahedron<> >() );
      this->basisType_         = BASIS_FEM_HIERARCHICAL;
      this->basisCoordinates_  = COORDINATES_CARTESIAN;
      this->functionSpace_     = FUNCTION_SPACE_HGRAD;
      
      const int degreeLength = 1;
      this->fieldOrdinalPolynomialDegree_ = OrdinalTypeArray2DHost("Integrated Legendre H(grad) tetrahedron polynomial degree lookup", this->basisCardinality_, degreeLength);
      
      int fieldOrdinalOffset = 0;
      // **** vertex functions **** //
      const int numVertices = this->basisCellTopology_.getVertexCount();
      const int numFunctionsPerVertex = 1;
      const int numVertexFunctions = numVertices * numFunctionsPerVertex;
      for (int i=0; i<numVertexFunctions; i++)
      {
        // for H(grad) on tetrahedron, if defineVertexFunctions is false, first four basis members are linear
        // if not, then the only difference is that the first member is constant
        this->fieldOrdinalPolynomialDegree_(i,0) = 1;
      }
      if (!defineVertexFunctions)
      {
        this->fieldOrdinalPolynomialDegree_(0,0) = 0;
      }
      fieldOrdinalOffset += numVertexFunctions;
      
      // **** edge functions **** //
      const int numFunctionsPerEdge = polyOrder - 1; // bubble functions: all but the vertices
      const int numEdges            = this->basisCellTopology_.getEdgeCount();
      for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
      {
        for (int i=0; i<numFunctionsPerEdge; i++)
        {
          this->fieldOrdinalPolynomialDegree_(i+fieldOrdinalOffset,0) = i+2; // vertex functions are 1st order; edge functions start at order 2
        }
        fieldOrdinalOffset += numFunctionsPerEdge;
      }
      
      // **** face functions **** //
      const int numFunctionsPerFace   = ((polyOrder-1)*(polyOrder-2))/2;
      const int numFaces = 4;
      for (int faceOrdinal=0; faceOrdinal<numFaces; faceOrdinal++)
      {
        for (int totalPolyOrder=3; totalPolyOrder<=polyOrder_; totalPolyOrder++)
        {
          const int totalFaceDofs         = (totalPolyOrder-2)*(totalPolyOrder-1)/2;
          const int totalFaceDofsPrevious = (totalPolyOrder-3)*(totalPolyOrder-2)/2;
          const int faceDofsForPolyOrder  = totalFaceDofs - totalFaceDofsPrevious;
          for (int i=0; i<faceDofsForPolyOrder; i++)
          {
            this->fieldOrdinalPolynomialDegree_(fieldOrdinalOffset,0) = totalPolyOrder;
            fieldOrdinalOffset++;
          }
        }
      }

      // **** interior functions **** //
      const int numFunctionsPerVolume = ((polyOrder-1)*(polyOrder-2)*(polyOrder-3))/6;
      const int numVolumes = 1; // interior
      for (int volumeOrdinal=0; volumeOrdinal<numVolumes; volumeOrdinal++)
      {
        for (int totalPolyOrder=4; totalPolyOrder<=polyOrder_; totalPolyOrder++)
        {
          const int totalInteriorDofs         = (totalPolyOrder-3)*(totalPolyOrder-2)*(totalPolyOrder-1)/6;
          const int totalInteriorDofsPrevious = (totalPolyOrder-4)*(totalPolyOrder-3)*(totalPolyOrder-2)/6;
          const int interiorDofsForPolyOrder  = totalInteriorDofs - totalInteriorDofsPrevious;
          
          for (int i=0; i<interiorDofsForPolyOrder; i++)
          {
            this->fieldOrdinalPolynomialDegree_(fieldOrdinalOffset,0) = totalPolyOrder;
            fieldOrdinalOffset++;
          }
        }
      }
      
      INTREPID2_TEST_FOR_EXCEPTION(fieldOrdinalOffset != this->basisCardinality_, std::invalid_argument, "Internal error: basis enumeration is incorrect");
      
      // initialize tags
      {
        // ESEAS numbers tetrahedron faces differently from Intrepid2
        // ESEAS:     012, 013, 123, 023
        // Intrepid2: 013, 123, 032, 021
        const int intrepid2FaceOrdinals[4] {3,0,1,2}; // index is the ESEAS face ordinal; value is the intrepid2 ordinal
        
        const auto & cardinality = this->basisCardinality_;
        
        // Basis-dependent initializations
        const ordinal_type tagSize  = 4;        // size of DoF tag, i.e., number of fields in the tag
        const ordinal_type posScDim = 0;        // position in the tag, counting from 0, of the subcell dim
        const ordinal_type posScOrd = 1;        // position in the tag, counting from 0, of the subcell ordinal
        const ordinal_type posDfOrd = 2;        // position in the tag, counting from 0, of DoF ordinal relative to the subcell
        
        OrdinalTypeArray1DHost tagView("tag view", cardinality*tagSize);
        const int vertexDim = 0, edgeDim = 1, faceDim = 2, volumeDim = 3;

        if (defineVertexFunctions) {
          {
            int tagNumber = 0;
            for (int vertexOrdinal=0; vertexOrdinal<numVertices; vertexOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerVertex; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = vertexDim;             // vertex dimension
                tagView(tagNumber*tagSize+1) = vertexOrdinal;         // vertex id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerVertex; // total number of dofs in this vertex
                tagNumber++;
              }
            }
            for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerEdge; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = edgeDim;               // edge dimension
                tagView(tagNumber*tagSize+1) = edgeOrdinal;           // edge id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerEdge;   // total number of dofs on this edge
                tagNumber++;
              }
            }
            for (int faceOrdinalESEAS=0; faceOrdinalESEAS<numFaces; faceOrdinalESEAS++)
            {
              int faceOrdinalIntrepid2 = intrepid2FaceOrdinals[faceOrdinalESEAS];
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerFace; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = faceDim;               // face dimension
                tagView(tagNumber*tagSize+1) = faceOrdinalIntrepid2;  // face id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerFace;   // total number of dofs on this face
                tagNumber++;
              }
            }
            for (int volumeOrdinal=0; volumeOrdinal<numVolumes; volumeOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerVolume; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = volumeDim;               // volume dimension
                tagView(tagNumber*tagSize+1) = volumeOrdinal;           // volume id
                tagView(tagNumber*tagSize+2) = functionOrdinal;         // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerVolume;   // total number of dofs in this volume
                tagNumber++;
              }
            }
            INTREPID2_TEST_FOR_EXCEPTION(tagNumber != this->basisCardinality_, std::invalid_argument, "Internal error: basis tag enumeration is incorrect");
          }
        } else {
          for (ordinal_type i=0;i<cardinality;++i) {
            tagView(i*tagSize+0) = volumeDim;   // volume dimension
            tagView(i*tagSize+1) = 0;           // volume ordinal
            tagView(i*tagSize+2) = i;           // local dof id
            tagView(i*tagSize+3) = cardinality; // total number of dofs on this face
          }
        }
        
        // Basis-independent function sets tag and enum data in tagToOrdinal_ and ordinalToTag_ arrays:
        // tags are constructed on host
        this->setOrdinalTagData(this->tagToOrdinal_,
                                this->ordinalToTag_,
                                tagView,
                                this->basisCardinality_,
                                tagSize,
                                posScDim,
                                posScOrd,
                                posDfOrd);
      }
    }
    
    const char* getName() const override {
      return "Intrepid2_IntegratedLegendreBasis_HGRAD_TET";
    }
    
    virtual bool requireOrientation() const override {
      return (this->getDegree() > 2);
    }

    // since the getValues() below only overrides the FEM variant, we specify that
    // we use the base class's getValues(), which implements the FVD variant by throwing an exception.
    // (It's an error to use the FVD variant on this basis.)
    using Basis<ExecutionSpace,OutputScalar,PointScalar>::getValues;
    
    virtual void getValues( OutputViewType outputValues, const PointViewType  inputPoints,
                           const EOperator operatorType = OPERATOR_VALUE ) const override
    {
      auto numPoints = inputPoints.extent_int(0);
      
      using FunctorType = Hierarchical_HGRAD_TET_Functor<ExecutionSpace, OutputScalar, PointScalar, OutputViewType, PointViewType>;
      
      FunctorType functor(operatorType, outputValues, inputPoints, polyOrder_, defineVertexFunctions);
      
      const int outputVectorSize = getVectorSizeForHierarchicalParallelism<OutputScalar>();
      const int pointVectorSize  = getVectorSizeForHierarchicalParallelism<PointScalar>();
      const int vectorSize = std::max(outputVectorSize,pointVectorSize);
      const int teamSize = 1; // because of the way the basis functions are computed, we don't have a second level of parallelism...
      
      auto policy = Kokkos::TeamPolicy<ExecutionSpace>(numPoints,teamSize,vectorSize);
      Kokkos::parallel_for( policy , functor, "Hierarchical_HGRAD_TET_Functor");
    }
  };
} // end namespace Intrepid2

#endif /* Intrepid2_IntegratedLegendreBasis_HGRAD_TET_h */