#include <hybridization.hpp>

Collaboration diagram for mfem::Hybridization:

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

	~Hybridization ()
	Destructor. More...

void	SetConstraintIntegrator (BilinearFormIntegrator *c_integ)

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

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

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

SparseMatrix &	GetMatrix ()
	Return the serial hybridized matrix. More...

HypreParMatrix &	GetParallelMatrix ()
	Return the parallel hybridized matrix. More...

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

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

Protected Member Functions
void	GetIBDofs (int el, Array< int > &i_dofs, Array< int > &b_dofs) const

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

double *	Af_data

int *	Af_ipiv

HypreParMatrix *	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,
has a left inverse , such that ,
a constraint matrix 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 64 of file hybridization.hpp.

Constructor & Destructor Documentation

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

Constructor.

Definition at line 30 of file hybridization.cpp.

mfem::Hybridization::~Hybridization ( )

Destructor.

Definition at line 40 of file hybridization.cpp.

Member Function Documentation

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

Assemble the element matrix A into the hybridized system matrix.

Definition at line 344 of file hybridization.cpp.

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 567 of file hybridization.cpp.

void mfem::Hybridization::Finalize ( )

Finalize the construction of the hybridized matrix.

Definition at line 434 of file hybridization.cpp.

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

protected

Definition at line 325 of file hybridization.cpp.

SparseMatrix& mfem::Hybridization::GetMatrix ( )

inline

Return the serial hybridized matrix.

Definition at line 115 of file hybridization.hpp.

HypreParMatrix& mfem::Hybridization::GetParallelMatrix ( )

inline

Return the parallel hybridized matrix.

Definition at line 119 of file hybridization.hpp.

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

Prepare the Hybridization object for assembly.

Definition at line 52 of file hybridization.cpp.

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

protected

Definition at line 456 of file hybridization.cpp.

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 538 of file hybridization.cpp.

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 102 of file hybridization.hpp.

Member Data Documentation

double* mfem::Hybridization::Af_data

protected

Definition at line 74 of file hybridization.hpp.

Array<int> mfem::Hybridization::Af_f_offsets

protected

Definition at line 73 of file hybridization.hpp.

int* mfem::Hybridization::Af_ipiv

protected

Definition at line 75 of file hybridization.hpp.

Array<int> mfem::Hybridization::Af_offsets

protected

Definition at line 73 of file hybridization.hpp.

BilinearFormIntegrator* mfem::Hybridization::c_bfi

protected

Definition at line 68 of file hybridization.hpp.

FiniteElementSpace * mfem::Hybridization::c_fes

protected

Definition at line 67 of file hybridization.hpp.

SparseMatrix* mfem::Hybridization::Ct

protected

Definition at line 70 of file hybridization.hpp.

FiniteElementSpace* mfem::Hybridization::fes

protected

Definition at line 67 of file hybridization.hpp.

SparseMatrix * mfem::Hybridization::H

protected

Definition at line 70 of file hybridization.hpp.

Array<int> mfem::Hybridization::hat_dofs_marker

protected

Definition at line 72 of file hybridization.hpp.

Array<int> mfem::Hybridization::hat_offsets

protected

Definition at line 72 of file hybridization.hpp.

HypreParMatrix* mfem::Hybridization::pH

protected

Definition at line 78 of file hybridization.hpp.

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

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation