Bramble-Pasciak Solver for Darcy equation. More...

#include <bramble_pasciak.hpp>

Inheritance diagram for mfem::blocksolvers::BramblePasciakSolver:

Collaboration diagram for mfem::blocksolvers::BramblePasciakSolver:

Public Member Functions
	BramblePasciakSolver (ParBilinearForm mVarf, ParMixedBilinearForm bVarf, const BPSParameters &param)
	System and mass preconditioner are constructed from bilinear forms.

	BramblePasciakSolver (HypreParMatrix &M, HypreParMatrix &B, HypreParMatrix &Q, Solver &M0, Solver &M1, const BPSParameters &param)
	System and mass preconditioner are user-provided.

virtual void	Mult (const Vector &x, Vector &y) const
	Operator application: `y=A(x)`.

virtual void	SetOperator (const Operator &op)
	Set/update the solver for the given operator.

void	SetEssZeroDofs (const Array< int > &dofs)

virtual int	GetNumIterations () const

Public Member Functions inherited from mfem::blocksolvers::DarcySolver
	DarcySolver (int size0, int size1)

Public Member Functions inherited from mfem::Solver
	Solver (int s=0, bool iter_mode=false)
	Initialize a square Solver with size s.

	Solver (int h, int w, bool iter_mode=false)
	Initialize a Solver with height h and width w.

Public Member Functions inherited from mfem::Operator
void	InitTVectors (const Operator Po, const Operator Ri, const Operator *Pi, Vector &x, Vector &b, Vector &X, Vector &B) const
	Initializes memory for true vectors of linear system.

	Operator (int s=0)
	Construct a square Operator with given size s (default 0).

	Operator (int h, int w)
	Construct an Operator with the given height (output size) and width (input size).

int	Height () const
	Get the height (size of output) of the Operator. Synonym with NumRows().

int	NumRows () const
	Get the number of rows (size of output) of the Operator. Synonym with Height().

int	Width () const
	Get the width (size of input) of the Operator. Synonym with NumCols().

int	NumCols () const
	Get the number of columns (size of input) of the Operator. Synonym with Width().

virtual MemoryClass	GetMemoryClass () const
	Return the MemoryClass preferred by the Operator.

virtual void	MultTranspose (const Vector &x, Vector &y) const
	Action of the transpose operator: `y=A^t(x)`. The default behavior in class Operator is to generate an error.

virtual void	AddMult (const Vector &x, Vector &y, const real_t a=1.0) const
	Operator application: `y+=A(x)` (default) or `y+=a*A(x)`.

virtual void	AddMultTranspose (const Vector &x, Vector &y, const real_t a=1.0) const
	Operator transpose application: `y+=A^t(x)` (default) or `y+=a*A^t(x)`.

virtual void	ArrayMult (const Array< const Vector * > &X, Array< Vector * > &Y) const
	Operator application on a matrix: `Y=A(X)`.

virtual void	ArrayMultTranspose (const Array< const Vector * > &X, Array< Vector * > &Y) const
	Action of the transpose operator on a matrix: `Y=A^t(X)`.

virtual void	ArrayAddMult (const Array< const Vector * > &X, Array< Vector * > &Y, const real_t a=1.0) const
	Operator application on a matrix: `Y+=A(X)` (default) or `Y+=a*A(X)`.

virtual void	ArrayAddMultTranspose (const Array< const Vector * > &X, Array< Vector * > &Y, const real_t a=1.0) const
	Operator transpose application on a matrix: `Y+=A^t(X)` (default) or `Y+=a*A^t(X)`.

virtual Operator &	GetGradient (const Vector &x) const
	Evaluate the gradient operator at the point x. The default behavior in class Operator is to generate an error.

virtual void	AssembleDiagonal (Vector &diag) const
	Computes the diagonal entries into diag. Typically, this operation only makes sense for linear Operators. In some cases, only an approximation of the diagonal is computed.

virtual const Operator *	GetProlongation () const
	Prolongation operator from linear algebra (linear system) vectors, to input vectors for the operator. `NULL` means identity.

virtual const Operator *	GetRestriction () const
	Restriction operator from input vectors for the operator to linear algebra (linear system) vectors. `NULL` means identity.

virtual const Operator *	GetOutputProlongation () const
	Prolongation operator from linear algebra (linear system) vectors, to output vectors for the operator. `NULL` means identity.

virtual const Operator *	GetOutputRestrictionTranspose () const
	Transpose of GetOutputRestriction, directly available in this form to facilitate matrix-free RAP-type operators.

virtual const Operator *	GetOutputRestriction () const
	Restriction operator from output vectors for the operator to linear algebra (linear system) vectors. `NULL` means identity.

void	FormLinearSystem (const Array< int > &ess_tdof_list, Vector &x, Vector &b, Operator *&A, Vector &X, Vector &B, int copy_interior=0)
	Form a constrained linear system using a matrix-free approach.

void	FormRectangularLinearSystem (const Array< int > &trial_tdof_list, const Array< int > &test_tdof_list, Vector &x, Vector &b, Operator *&A, Vector &X, Vector &B)
	Form a column-constrained linear system using a matrix-free approach.

virtual void	RecoverFEMSolution (const Vector &X, const Vector &b, Vector &x)
	Reconstruct a solution vector x (e.g. a GridFunction) from the solution X of a constrained linear system obtained from Operator::FormLinearSystem() or Operator::FormRectangularLinearSystem().

void	FormSystemOperator (const Array< int > &ess_tdof_list, Operator *&A)
	Return in A a parallel (on truedofs) version of this square operator.

void	FormRectangularSystemOperator (const Array< int > &trial_tdof_list, const Array< int > &test_tdof_list, Operator *&A)
	Return in A a parallel (on truedofs) version of this rectangular operator (including constraints).

void	FormDiscreteOperator (Operator *&A)
	Return in A a parallel (on truedofs) version of this rectangular operator.

void	PrintMatlab (std::ostream &out, int n, int m=0) const
	Prints operator with input size n and output size m in Matlab format.

virtual void	PrintMatlab (std::ostream &out) const
	Prints operator in Matlab format.

virtual	~Operator ()
	Virtual destructor.

Type	GetType () const
	Return the type ID of the Operator class.

Static Public Member Functions
static HypreParMatrix *	ConstructMassPreconditioner (ParBilinearForm &mVarf, real_t alpha=0.5)
	Assemble a preconditioner for the mass matrix.

Additional Inherited Members
Public Types inherited from mfem::Operator
enum	DiagonalPolicy { DIAG_ZERO , DIAG_ONE , DIAG_KEEP }
	Defines operator diagonal policy upon elimination of rows and/or columns. More...

enum	Type { ANY_TYPE , MFEM_SPARSEMAT , Hypre_ParCSR , PETSC_MATAIJ , PETSC_MATIS , PETSC_MATSHELL , PETSC_MATNEST , PETSC_MATHYPRE , PETSC_MATGENERIC , Complex_Operator , MFEM_ComplexSparseMat , Complex_Hypre_ParCSR , Complex_DenseMat , MFEM_Block_Matrix , MFEM_Block_Operator }
	Enumeration defining IDs for some classes derived from Operator. More...

Public Attributes inherited from mfem::Solver
bool	iterative_mode
	If true, use the second argument of Mult() as an initial guess.

Protected Member Functions inherited from mfem::Operator
void	FormConstrainedSystemOperator (const Array< int > &ess_tdof_list, ConstrainedOperator *&Aout)
	see FormSystemOperator()

void	FormRectangularConstrainedSystemOperator (const Array< int > &trial_tdof_list, const Array< int > &test_tdof_list, RectangularConstrainedOperator *&Aout)
	see FormRectangularSystemOperator()

Operator *	SetupRAP (const Operator Pi, const Operator Po)
	Returns RAP Operator of this, using input/output Prolongation matrices Pi corresponds to "P", Po corresponds to "Rt".

Protected Attributes inherited from mfem::blocksolvers::DarcySolver
Array< int >	offsets_

Protected Attributes inherited from mfem::Operator
int	height
	Dimension of the output / number of rows in the matrix.

int	width
	Dimension of the input / number of columns in the matrix.

Detailed Description

Bramble-Pasciak Solver for Darcy equation.

The basic idea is to precondition the mass matrix M with a s.p.d. matrix Q such that M - Q remains s.p.d. Then we can transform the block operator into a s.p.d. operator under a modified inner product. In particular, this enable us to implement modified versions of CG iterations, that rely on efficient applications of the required transformations.

We offer a mass preconditioner based on a rescaling of the diagonal of the element mass matrices M_T.

We consider Q_T := alpha * lambda_min * D_T, where D_T := diag(M_T), and lambda_min is the smallest eigenvalue of the following problem

               M_T x = lambda * D_T x.

Alpha is a parameter that is strictly between 0 and 1.

For more details, see:

P. Vassilevski, Multilevel Block Factorization Preconditioners (Appendix F.3), Springer, 2008.
J. Bramble and J. Pasciak. A Preconditioning Technique for Indefinite Systems Resulting From Mixed Approximations of Elliptic Problems, Mathematics of Computation, 50:1-17, 1988.

Definition at line 122 of file bramble_pasciak.hpp.

Constructor & Destructor Documentation

◆ BramblePasciakSolver() [1/2]

mfem::blocksolvers::BramblePasciakSolver::BramblePasciakSolver	(	ParBilinearForm *	mVarf,
		ParMixedBilinearForm *	bVarf,
		const BPSParameters &	param )

System and mass preconditioner are constructed from bilinear forms.

Bramble-Pasciak Solver.

Definition at line 22 of file bramble_pasciak.cpp.

◆ BramblePasciakSolver() [2/2]

mfem::blocksolvers::BramblePasciakSolver::BramblePasciakSolver	(	HypreParMatrix &	M,
		HypreParMatrix &	B,
		HypreParMatrix &	Q,
		Solver &	M0,
		Solver &	M1,
		const BPSParameters &	param )

System and mass preconditioner are user-provided.

Definition at line 46 of file bramble_pasciak.cpp.

Member Function Documentation

◆ ConstructMassPreconditioner()

HypreParMatrix * mfem::blocksolvers::BramblePasciakSolver::ConstructMassPreconditioner	(	ParBilinearForm &	mVarf,
		real_t	alpha = 0.5 )

static

Assemble a preconditioner for the mass matrix.

Mass preconditioner corresponds to a local re-scaling based on the smallest eigenvalue of the generalized eigenvalue problem locally on each element T: M_T x_T = lambda_T diag(M_T) x_T. We set Q_T = alpha * min(lambda_T) * diag(M_T), 0 < alpha < 1.

Definition at line 135 of file bramble_pasciak.cpp.

◆ GetNumIterations()

virtual int mfem::blocksolvers::BramblePasciakSolver::GetNumIterations ( ) const

inlinevirtual

Implements mfem::blocksolvers::DarcySolver.

Definition at line 167 of file bramble_pasciak.hpp.

◆ Mult()

void mfem::blocksolvers::BramblePasciakSolver::Mult	(	const Vector &	x,
		Vector &	y ) const

virtual

Operator application: y=A(x).

Implements mfem::Operator.

Definition at line 207 of file bramble_pasciak.cpp.

◆ SetEssZeroDofs()

void mfem::blocksolvers::BramblePasciakSolver::SetEssZeroDofs ( const Array< int > & dofs )

inline

Definition at line 166 of file bramble_pasciak.hpp.

◆ SetOperator()

virtual void mfem::blocksolvers::BramblePasciakSolver::SetOperator ( const Operator & op )

inlinevirtual

Set/update the solver for the given operator.

Implements mfem::Solver.

Definition at line 165 of file bramble_pasciak.hpp.

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

miniapps/solvers/bramble_pasciak.hpp
miniapps/solvers/bramble_pasciak.cpp

Public Member Functions

Static Public Member Functions

Additional Inherited Members