hybridization.hpp

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// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.

#ifndef MFEM_HYBRIDIZATION
#define MFEM_HYBRIDIZATION

#include "../config/config.hpp"
#include "fespace.hpp"
#include "bilininteg.hpp"

namespace mfem
{

/** Auxiliary class Hybridization, used to implement BilinearForm hybridization.

    Hybridization can be viewed as a technique for solving linear systems
    obtained through finite element assembly. The assembled matrix \f$ A \f$ can
    be written as:
        \f[ A = P^T \hat{A} P, \f]
    where \f$ P \f$ is the matrix mapping the conforming finite element space to
    the purely local finite element space without any inter-element constraints
    imposed, and \f$ \hat{A} \f$ is the block-diagonal matrix of all element
    matrices.

    We assume that:
    - \f$ \hat{A} \f$ is invertible,
    - \f$ P \f$ has a left inverse \f$ R \f$, such that \f$ R P = I \f$,
    - a constraint matrix \f$ C \f$ can be constructed, such that
      \f$ \operatorname{Ker}(C) = \operatorname{Im}(P) \f$.

    Under these conditions, the linear system \f$ A x = b \f$ can be solved
    using the following procedure:
    - solve for \f$ \lambda \f$ in the linear system:
          \f[ (C \hat{A}^{-1} C^T) \lambda = C \hat{A}^{-1} R^T b \f]
    - compute \f$ x = R \hat{A}^{-1} (R^T b - C^T \lambda) \f$

    Hybridization is advantageous when the matrix
    \f$ H = (C \hat{A}^{-1} C^T) \f$ of the hybridized system is either smaller
    than the original system, or is simpler to invert with a known method.

    In some cases, e.g. high-order elements, the matrix \f$ C \f$ can be written
    as
        \f[ C = \begin{pmatrix} 0 & C_b \end{pmatrix}, \f]
    and then the hybridized matrix \f$ H \f$ can be assembled using the identity
        \f[ H = C_b S_b^{-1} C_b^T, \f]
    where \f$ S_b \f$ is the Schur complement of \f$ \hat{A} \f$ with respect to
    the same decomposition as the columns of \f$ C \f$:
        \f[ S_b = \hat{A}_b - \hat{A}_{bf} \hat{A}_{f}^{-1} \hat{A}_{fb}. \f]

    Hybridization can also be viewed as a discretization method for imposing
    (weak) continuity constraints between neighboring elements. */
class Hybridization
{
protected:
   FiniteElementSpace *fes, *c_fes;
   BilinearFormIntegrator *c_bfi;

   SparseMatrix *Ct, *H;

   Array<int> hat_offsets, hat_dofs_marker;
   Array<int> Af_offsets, Af_f_offsets;
   double *Af_data;
   int *Af_ipiv;

#ifdef MFEM_USE_MPI
   HypreParMatrix *pC, *P_pc; // for parallel non-conforming meshes
   OperatorHandle pH;
#endif

   void ConstructC();

   void GetIBDofs(int el, Array<int> &i_dofs, Array<int> &b_dofs) const;

   void GetBDofs(int el, int &num_idofs, Array<int> &b_dofs) const;

   void ComputeH();

   // Compute depending on mode:
   // - mode 0: bf = Af^{-1} Rf^t b, where
   //           the non-"boundary" part of bf is set to 0;
   // - mode 1: bf = Af^{-1} ( Rf^t b - Cf^t lambda ), where
   //           the "essential" part of bf is set to 0.
   // Input: size(b)      =   fes->GetConformingVSize()
   //        size(lambda) = c_fes->GetConformingVSize()
   void MultAfInv(const Vector &b, const Vector &lambda, Vector &bf,
                  int mode) const;

public:
   /// Constructor
   Hybridization(FiniteElementSpace *fespace, FiniteElementSpace *c_fespace);
   /// Destructor
   ~Hybridization();

   /** Set the integrator that will be used to construct the constraint matrix
       C. The Hybridization object assumes ownership of the integrator, i.e. it
       will delete the integrator when destroyed. */
   void SetConstraintIntegrator(BilinearFormIntegrator *c_integ)
   { delete c_bfi; c_bfi = c_integ; }

   /// Prepare the Hybridization object for assembly.
   void Init(const Array<int> &ess_tdof_list);

   /// Assemble the element matrix A into the hybridized system matrix.
   void AssembleMatrix(int el, const DenseMatrix &A);

   /// Assemble the boundary element matrix A into the hybridized system matrix.
   void AssembleBdrMatrix(int bdr_el, const DenseMatrix &A);

   /// Finalize the construction of the hybridized matrix.
   void Finalize();

   /// Return the serial hybridized matrix.
   SparseMatrix &GetMatrix() { return *H; }

#ifdef MFEM_USE_MPI
   /// Return the parallel hybridized matrix.
   HypreParMatrix &GetParallelMatrix() { return *pH.Is<HypreParMatrix>(); }

   /** @brief Return the parallel hybridized matrix in the format specified by
       SetOperatorType(). */
   void GetParallelMatrix(OperatorHandle &H_h) const { H_h = pH; }

   /// Set the operator type id for the parallel hybridized matrix/operator.
   void SetOperatorType(Operator::Type tid) { pH.SetType(tid); }
#endif

   /** Perform the reduction of the given r.h.s. vector, b, to a r.h.s vector,
       b_r, for the hybridized system. */
   void ReduceRHS(const Vector &b, Vector &b_r) const;

   /** Reconstruct the solution of the original system, sol, from solution of
       the hybridized system, sol_r, and the original r.h.s. vector, b.
       It is assumed that the vector sol has the right essential b.c. */
   void ComputeSolution(const Vector &b, const Vector &sol_r,
                        Vector &sol) const;

   /** @brief Destroy the current hybridization matrix while preserving the
       computed constraint matrix and the set of essential true dofs. After
       Reset(), a new hybridized matrix can be assembled via AssembleMatrix()
       and Finalize(). The Mesh and FiniteElementSpace objects are assumed to be
       un-modified. If that is not the case, a new Hybridization object must be
       created. */
   void Reset();
};

}

#endif