#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	AssembleBdrMatrix (int bdr_el, const DenseMatrix &A)
	Assemble the boundary 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

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. More...

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

double *	Af_data

int *	Af_ipiv

HypreParMatrix *	pC

HypreParMatrix *	P_pc

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 41 of file hybridization.cpp.

Member Function Documentation

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

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

Definition at line 481 of file hybridization.cpp.

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

Assemble the element matrix A into the hybridized system matrix.

Definition at line 441 of file hybridization.cpp.

void mfem::Hybridization::ComputeH ( )

protected

Definition at line 569 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 823 of file hybridization.cpp.

void mfem::Hybridization::ConstructC ( )

protected

Definition at line 55 of file hybridization.cpp.

void mfem::Hybridization::Finalize ( )

Finalize the construction of the hybridized matrix.

Definition at line 706 of file hybridization.cpp.

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

protected

Definition at line 425 of file hybridization.cpp.

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

protected

Definition at line 405 of file hybridization.cpp.

SparseMatrix& mfem::Hybridization::GetMatrix ( )

inline

Return the serial hybridized matrix.

Definition at line 125 of file hybridization.hpp.

HypreParMatrix& mfem::Hybridization::GetParallelMatrix ( )

inline

Return the parallel hybridized matrix.

Definition at line 129 of file hybridization.hpp.

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

Prepare the Hybridization object for assembly.

Definition at line 214 of file hybridization.cpp.

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

protected

Definition at line 715 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 796 of file hybridization.cpp.

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 862 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 109 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::P_pc

protected

Definition at line 78 of file hybridization.hpp.

HypreParMatrix* mfem::Hybridization::pC

protected

Definition at line 78 of file hybridization.hpp.

HypreParMatrix* mfem::Hybridization::pH

protected

Definition at line 79 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