Auxiliary class Hybridization, used to implement BilinearForm hybridization. More...

#include <hybridization.hpp>

Collaboration diagram for mfem::Hybridization:

Public Member Functions
	Hybridization (FiniteElementSpace fespace, FiniteElementSpace c_fespace)
	Constructor.

	~Hybridization ()
	Destructor.

void	SetConstraintIntegrator (BilinearFormIntegrator *c_integ)

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

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

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

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

SparseMatrix &	GetMatrix ()
	Return the serial hybridized matrix.

HypreParMatrix &	GetParallelMatrix ()
	Return the parallel hybridized matrix.

void	GetParallelMatrix (OperatorHandle &H_h) const
	Return the parallel hybridized matrix in the format specified by SetOperatorType().

void	SetOperatorType (Operator::Type tid)
	Set the operator type id for the parallel hybridized matrix/operator.

void	ReduceRHS (const Vector &b, Vector &b_r) const

void	ComputeSolution (const Vector &b, const Vector &sol_r, Vector &sol) const

void	Reset ()
	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.

Protected Member Functions
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 ()

void	MultAfInv (const Vector &b, const Vector &lambda, Vector &bf, int mode) const

Protected Attributes
FiniteElementSpace *	fes

FiniteElementSpace *	c_fes

BilinearFormIntegrator *	c_bfi

SparseMatrix *	Ct

SparseMatrix *	H

Array< int >	hat_offsets

Array< int >	hat_dofs_marker

Array< int >	Af_offsets

Array< int >	Af_f_offsets

real_t *	Af_data

int *	Af_ipiv

HypreParMatrix *	pC

HypreParMatrix *	P_pc

OperatorHandle	pH

Detailed Description

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 \( A \) can be written as:

\[ A = P^T \hat{A} P, \]

where \( P \) is the matrix mapping the conforming finite element space to the purely local finite element space without any inter-element constraints imposed, and \( \hat{A} \) is the block-diagonal matrix of all element matrices.

We assume that:

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

Under these conditions, the linear system \( A x = b \) can be solved using the following procedure:

solve for \( \lambda \) in the linear system:
\[ (C \hat{A}^{-1} C^T) \lambda = C \hat{A}^{-1} R^T b \]
compute \( x = R \hat{A}^{-1} (R^T b - C^T \lambda) \)

Hybridization is advantageous when the matrix \( H = (C \hat{A}^{-1} C^T) \) 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 \( C \) can be written as

\[ C = \begin{pmatrix} 0 & C_b \end{pmatrix}, \]

and then the hybridized matrix \( H \) can be assembled using the identity

\[ H = C_b S_b^{-1} C_b^T, \]

where \( S_b \) is the Schur complement of \( \hat{A} \) with respect to the same decomposition as the columns of \( C \):

\[ S_b = \hat{A}_b - \hat{A}_{bf} \hat{A}_{f}^{-1} \hat{A}_{fb}. \]

Hybridization can also be viewed as a discretization method for imposing (weak) continuity constraints between neighboring elements.

Definition at line 61 of file hybridization.hpp.

Constructor & Destructor Documentation

◆ Hybridization()

mfem::Hybridization::Hybridization	(	FiniteElementSpace *	fespace,
		FiniteElementSpace *	c_fespace )

Constructor.

Definition at line 30 of file hybridization.cpp.

◆ ~Hybridization()

mfem::Hybridization::~Hybridization ( )

Destructor.

Definition at line 41 of file hybridization.cpp.

Member Function Documentation

◆ AssembleBdrMatrix()

void mfem::Hybridization::AssembleBdrMatrix	(	int	bdr_el,
		const DenseMatrix &	A )

Assemble the boundary element matrix A into the hybridized system matrix.

Definition at line 482 of file hybridization.cpp.

◆ AssembleMatrix()

void mfem::Hybridization::AssembleMatrix	(	int	el,
		const DenseMatrix &	A )

Assemble the element matrix A into the hybridized system matrix.

Definition at line 442 of file hybridization.cpp.

◆ ComputeH()

void mfem::Hybridization::ComputeH ( )

protected

Definition at line 570 of file hybridization.cpp.

◆ ComputeSolution()

void mfem::Hybridization::ComputeSolution	(	const Vector &	b,
		const Vector &	sol_r,
		Vector &	sol ) 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.

Definition at line 841 of file hybridization.cpp.

◆ ConstructC()

void mfem::Hybridization::ConstructC ( )

protected

Definition at line 54 of file hybridization.cpp.

◆ Finalize()

void mfem::Hybridization::Finalize ( )

Finalize the construction of the hybridized matrix.

Definition at line 717 of file hybridization.cpp.

◆ GetBDofs()

void mfem::Hybridization::GetBDofs	(	int	el,
		int &	num_idofs,
		Array< int > &	b_dofs ) const

protected

Definition at line 426 of file hybridization.cpp.

◆ GetIBDofs()

void mfem::Hybridization::GetIBDofs	(	int	el,
		Array< int > &	i_dofs,
		Array< int > &	b_dofs ) const

protected

Definition at line 406 of file hybridization.cpp.

◆ GetMatrix()

SparseMatrix & mfem::Hybridization::GetMatrix ( )

inline

Return the serial hybridized matrix.

Definition at line 122 of file hybridization.hpp.

◆ GetParallelMatrix() [1/2]

HypreParMatrix & mfem::Hybridization::GetParallelMatrix ( )

inline

Return the parallel hybridized matrix.

Definition at line 126 of file hybridization.hpp.

◆ GetParallelMatrix() [2/2]

void mfem::Hybridization::GetParallelMatrix ( OperatorHandle & H_h ) const

inline

Return the parallel hybridized matrix in the format specified by SetOperatorType().

Definition at line 130 of file hybridization.hpp.

◆ Init()

void mfem::Hybridization::Init ( const Array< int > & ess_tdof_list )

Prepare the Hybridization object for assembly.

Definition at line 215 of file hybridization.cpp.

◆ MultAfInv()

void mfem::Hybridization::MultAfInv	(	const Vector &	b,
		const Vector &	lambda,
		Vector &	bf,
		int	mode ) const

protected

Definition at line 726 of file hybridization.cpp.

◆ ReduceRHS()

void mfem::Hybridization::ReduceRHS	(	const Vector &	b,
		Vector &	b_r ) const

Perform the reduction of the given r.h.s. vector, b, to a r.h.s vector, b_r, for the hybridized system.

Definition at line 807 of file hybridization.cpp.

◆ Reset()

void mfem::Hybridization::Reset ( )

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.

Definition at line 880 of file hybridization.cpp.

◆ SetConstraintIntegrator()

void mfem::Hybridization::SetConstraintIntegrator ( BilinearFormIntegrator * c_integ )

inline

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.

Definition at line 106 of file hybridization.hpp.

◆ SetOperatorType()

void mfem::Hybridization::SetOperatorType ( Operator::Type tid )

inline

Set the operator type id for the parallel hybridized matrix/operator.

Definition at line 133 of file hybridization.hpp.

Member Data Documentation

◆ Af_data

real_t* mfem::Hybridization::Af_data

protected

Definition at line 71 of file hybridization.hpp.

◆ Af_f_offsets

Array<int> mfem::Hybridization::Af_f_offsets

protected

Definition at line 70 of file hybridization.hpp.

◆ Af_ipiv

int* mfem::Hybridization::Af_ipiv

protected

Definition at line 72 of file hybridization.hpp.

◆ Af_offsets

Array<int> mfem::Hybridization::Af_offsets

protected

Definition at line 70 of file hybridization.hpp.

◆ c_bfi

BilinearFormIntegrator* mfem::Hybridization::c_bfi

protected

Definition at line 65 of file hybridization.hpp.

◆ c_fes

FiniteElementSpace * mfem::Hybridization::c_fes

protected

Definition at line 64 of file hybridization.hpp.

◆ Ct

SparseMatrix* mfem::Hybridization::Ct

protected

Definition at line 67 of file hybridization.hpp.

◆ fes

FiniteElementSpace* mfem::Hybridization::fes

protected

Definition at line 64 of file hybridization.hpp.

◆ H

SparseMatrix * mfem::Hybridization::H

protected

Definition at line 67 of file hybridization.hpp.

◆ hat_dofs_marker

Array<int> mfem::Hybridization::hat_dofs_marker

protected

Definition at line 69 of file hybridization.hpp.

◆ hat_offsets

Array<int> mfem::Hybridization::hat_offsets

protected

Definition at line 69 of file hybridization.hpp.

◆ P_pc

HypreParMatrix * mfem::Hybridization::P_pc

protected

Definition at line 75 of file hybridization.hpp.

◆ pC

HypreParMatrix* mfem::Hybridization::pC

protected

Definition at line 75 of file hybridization.hpp.

◆ pH

OperatorHandle mfem::Hybridization::pH

protected

Definition at line 76 of file hybridization.hpp.

The documentation for this class was generated from the following files:

fem/hybridization.hpp
fem/hybridization.cpp

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Hybridization()

◆ ~Hybridization()

Member Function Documentation

◆ AssembleBdrMatrix()

◆ AssembleMatrix()

◆ ComputeH()

◆ ComputeSolution()

◆ ConstructC()

◆ Finalize()

◆ GetBDofs()

◆ GetIBDofs()

◆ GetMatrix()

◆ GetParallelMatrix() [1/2]

◆ GetParallelMatrix() [2/2]

◆ Init()

◆ MultAfInv()

◆ ReduceRHS()

◆ Reset()

◆ SetConstraintIntegrator()

◆ SetOperatorType()

Member Data Documentation

◆ Af_data

◆ Af_f_offsets

◆ Af_ipiv

◆ Af_offsets

◆ c_bfi

◆ c_fes

◆ Ct

◆ fes

◆ H

◆ hat_dofs_marker

◆ hat_offsets

◆ P_pc

◆ pC

◆ pH