Uzawa preconditioner. More...

#include <uzawa_precond.hpp>

Inheritance diagram for FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >:

Public Types
typedef SolverBase< VectorType >	BaseClass
	base-class typedef More...

typedef MatrixA_::DataType	DataType
	our data type More...

typedef SolverBase< VectorTypeV >	SolverA
	A-block solver type. More...

typedef SolverBase< VectorTypeP >	SolverS
	S-block solver type. More...

typedef LAFEM::TupleVector< VectorTypeV, VectorTypeP >	VectorType
	system vector type More...

typedef MatrixD_::VectorTypeL	VectorTypeP
	pressure vector type More...

typedef MatrixB_::VectorTypeL	VectorTypeV
	velocity vector type More...

Public Member Functions
	UzawaPrecond (const MatrixA_ &matrix_a, const MatrixB_ &matrix_b, const MatrixD_ &matrix_d, const FilterV_ &filter_v, const FilterP_ &filter_p, std::shared_ptr< SolverA > solver_a, std::shared_ptr< SolverS > solver_s, UzawaType type=UzawaType::diagonal, bool auto_init_s=true)
	Constructs a Uzawa preconditioner. More...

virtual Status	apply (LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > &vec_cor, const LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > &vec_def)=0
	Solver application method. More...

virtual Status	apply (VectorType &vec_cor, const VectorType &vec_def) override

virtual void	done ()
	Finalization method. More...

virtual void	done_numeric () override
	Numeric finalization method. More...

virtual void	done_symbolic () override
	Symbolic finalization method. More...

virtual void	init ()
	Initialization method. More...

virtual void	init_numeric () override
	Numeric initialization method. More...

virtual void	init_symbolic () override
	Symbolic initialization method. More...

virtual String	name () const override
	Returns a descriptive string. More...

Private Attributes
const bool	_auto_init_s
	auto-initialize of S-solver More...

const FilterP_ &	_filter_p

const FilterV_ &	_filter_v
	our system filter More...

const MatrixA_ &	_matrix_a

const MatrixB_ &	_matrix_b

const MatrixD_ &	_matrix_d

std::shared_ptr< SolverA >	_solver_a
	our A-block solver More...

std::shared_ptr< SolverS >	_solver_s
	our S-block solver More...

UzawaType	_uzawa_type
	our Uzawa type More...

VectorTypeP	_vec_tmp_p

VectorTypeV	_vec_tmp_v
	a temporary defect vector More...

Detailed Description

template<typename MatrixA_, typename MatrixB_, typename MatrixD_, typename FilterV_, typename FilterP_>
class FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >

Uzawa preconditioner.

This class implements the Uzawa preconditioner, which is a special preconditioner for saddle-point systems of the form

\[\begin{bmatrix} A & B\\D & 0\end{bmatrix} \cdot \begin{bmatrix} x_u\\x_p\end{bmatrix} = \begin{bmatrix} f_u\\f_p\end{bmatrix}\]

Let \( S \approx -DA^{-1}B\) be an approximation of the Schur-complement matrix, then this class supports a total of four different preconditioners for the saddle-point system above:

UzawaType::diagonal: Solves the system
\[\begin{bmatrix} A & 0\\0 & S\end{bmatrix} \cdot \begin{bmatrix} x_u\\x_p\end{bmatrix} = \begin{bmatrix} f_u\\f_p\end{bmatrix}\]
UzawaType::lower: Solves the system
\[\begin{bmatrix} A & 0\\D & S\end{bmatrix} \cdot \begin{bmatrix} x_u\\x_p\end{bmatrix} = \begin{bmatrix} f_u\\f_p\end{bmatrix}\]
UzawaType::upper: Solves the system
\[\begin{bmatrix} A & B\\0 & S\end{bmatrix} \cdot \begin{bmatrix} x_u\\x_p\end{bmatrix} = \begin{bmatrix} f_u\\f_p\end{bmatrix}\]
UzawaType::full: Solves the system
\[\begin{bmatrix} I & 0\\-DA^{-1} & I\end{bmatrix} \cdot \begin{bmatrix} A & B\\0 & S\end{bmatrix} \cdot \begin{bmatrix} x_u\\x_p\end{bmatrix} = \begin{bmatrix} f_u\\f_p\end{bmatrix}\]

The required solution steps of \( A^{-1} \) and \( S^{-1} \) are performed by two sub-solvers, which have to be created by the user and supplied to the constructor of this object.

Template Parameters

MatrixA_,MatrixB_,MatrixD_	The types of the sub-matrices A, B and D.
FilterV_,FilterP_	The types of the filters for the velocity and pressure components, respectively.

Note: This class template is specialized for Global::Matrix and Global::Filter instances below.

Author: Peter Zajac

Definition at line 76 of file uzawa_precond.hpp.

Member Typedef Documentation

◆ BaseClass

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef SolverBase<VectorType> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::BaseClass

base-class typedef

Definition at line 89 of file uzawa_precond.hpp.

◆ DataType

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef MatrixA_::DataType FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::DataType

our data type

Definition at line 81 of file uzawa_precond.hpp.

◆ SolverA

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef SolverBase<VectorTypeV> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::SolverA

A-block solver type.

Definition at line 92 of file uzawa_precond.hpp.

◆ SolverS

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef SolverBase<VectorTypeP> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::SolverS

S-block solver type.

Definition at line 94 of file uzawa_precond.hpp.

◆ VectorType

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef LAFEM::TupleVector<VectorTypeV, VectorTypeP> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::VectorType

system vector type

Definition at line 87 of file uzawa_precond.hpp.

◆ VectorTypeP

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef MatrixD_::VectorTypeL FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::VectorTypeP

pressure vector type

Definition at line 85 of file uzawa_precond.hpp.

◆ VectorTypeV

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

typedef MatrixB_::VectorTypeL FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::VectorTypeV

velocity vector type

Definition at line 83 of file uzawa_precond.hpp.

Constructor & Destructor Documentation

◆ UzawaPrecond()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::UzawaPrecond	(	const MatrixA_ &	matrix_a,
		const MatrixB_ &	matrix_b,
		const MatrixD_ &	matrix_d,
		const FilterV_ &	filter_v,
		const FilterP_ &	filter_p,
		std::shared_ptr< SolverA >	solver_a,
		std::shared_ptr< SolverS >	solver_s,
		UzawaType	type = `UzawaType::diagonal`,
		bool	auto_init_s = `true`
	)

inlineexplicit

Constructs a Uzawa preconditioner.

Parameters

[in]	matrix_a,matrix_b,matrix_d	The three sub-matrices of the saddle-point matrix.
[in]	filter_v,filter_p	The velocity and pressure filter, resp.
[in]	solver_a	The solver representing \(A^{-1}\).
[in]	solver_s	The solver representing \(S^{-1}\).
[in]	type	Specifies the type of the preconditioner. See this class' documentation for details.
[in]	auto_init_s	Specifies whether this solver object should call the init/done functions of the `solver_s` object. If set to `false`, then the caller is responsible for the initialization and finalization of the S-block solver object.

Definition at line 139 of file uzawa_precond.hpp.

References XASSERTM.

◆ ~UzawaPrecond()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::~UzawaPrecond ( )

inlinevirtual

Definition at line 163 of file uzawa_precond.hpp.

Member Function Documentation

◆ apply() [1/2]

virtual Status FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >::apply	(	LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > &	vec_cor,
		const LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > &	vec_def
	)

pure virtualinherited

Solver application method.

This method applies the solver represented by this object onto a given defect vector and returns the corresponding correction vector.

Note: Solvers which derive from the IterativeSolver base class also provide a correct() method which corrects an initial solution instead of starting with the null vector.

Parameters

[out]	vec_cor	The vector that shall receive the solution of the linear system. It is assumed to be allocated, but its numerical contents may be undefined upon calling this method.
[in]	vec_def	The vector that represents the right-hand-side of the linear system to be solved.

Attention: vec_cor and vec_def must not refer to the same vector object!

Returns: A Status code that represents the status of the solution step.

◆ apply() [2/2]

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual Status FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::apply	(	VectorType &	vec_cor,
		const VectorType &	vec_def
	)

inlineoverridevirtual

Definition at line 224 of file uzawa_precond.hpp.

◆ done()

virtual void FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >::done ( )

inlinevirtualinherited

Finalization method.

This function performs both the symbolic and numeric finalization, i.e. it simply performs

this->done_numeric();
this->done_symbolic();

Definition at line 283 of file base.hpp.

◆ done_numeric()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual void FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::done_numeric ( )

inlineoverridevirtual

Numeric finalization method.

This method is called to release any data allocated in the numeric initialization step.

Reimplemented from FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >.

Definition at line 199 of file uzawa_precond.hpp.

References FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_auto_init_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_a, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_s, and FEAT::Solver::SolverBase< Vector_ >::done_numeric().

◆ done_symbolic()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual void FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::done_symbolic ( )

inlineoverridevirtual

Symbolic finalization method.

This method is called to release any data allocated in the symbolic initialization step.

Reimplemented from FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >.

Definition at line 209 of file uzawa_precond.hpp.

References FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_auto_init_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_a, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_uzawa_type, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_vec_tmp_v, FEAT::Solver::diagonal, and FEAT::Solver::SolverBase< Vector_ >::done_symbolic().

◆ init()

virtual void FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >::init ( )

inlinevirtualinherited

Initialization method.

This function performs both the symbolic and numeric initialization, i.e. it simply performs

this->init_symbolic();
this->init_numeric();

Definition at line 268 of file base.hpp.

◆ init_numeric()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual void FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::init_numeric ( )

inlineoverridevirtual

Numeric initialization method.

This method is called to perform numeric initialization of the solver.
Before this function can be called, the symbolic initialization must be performed.

Reimplemented from FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >.

Definition at line 189 of file uzawa_precond.hpp.

References FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_auto_init_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_a, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_s, and FEAT::Solver::SolverBase< Vector_ >::init_numeric().

◆ init_symbolic()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual void FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::init_symbolic ( )

inlineoverridevirtual

Symbolic initialization method.

This method is called to perform symbolic initialization of the solver.

Reimplemented from FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >.

Definition at line 172 of file uzawa_precond.hpp.

References FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_auto_init_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_a, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_s, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_uzawa_type, FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_vec_tmp_v, FEAT::Solver::diagonal, and FEAT::Solver::SolverBase< Vector_ >::init_symbolic().

◆ name()

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

virtual String FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::name ( ) const

inlineoverridevirtual

Returns a descriptive string.

Returns: A string describing the solver.

Implements FEAT::Solver::SolverBase< LAFEM::TupleVector< MatrixB_::VectorTypeL, MatrixD_::VectorTypeL > >.

Definition at line 167 of file uzawa_precond.hpp.

Member Data Documentation

◆ _auto_init_s

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const bool FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_auto_init_s

private

auto-initialize of S-solver

Definition at line 110 of file uzawa_precond.hpp.

◆ _filter_p

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const FilterP_& FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_filter_p

private

Definition at line 102 of file uzawa_precond.hpp.

◆ _filter_v

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const FilterV_& FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_filter_v

private

our system filter

Definition at line 101 of file uzawa_precond.hpp.

◆ _matrix_a

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const MatrixA_& FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_matrix_a

private

Definition at line 97 of file uzawa_precond.hpp.

◆ _matrix_b

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const MatrixB_& FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_matrix_b

private

Definition at line 98 of file uzawa_precond.hpp.

◆ _matrix_d

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

const MatrixD_& FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_matrix_d

private

Definition at line 99 of file uzawa_precond.hpp.

◆ _solver_a

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

std::shared_ptr<SolverA> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_a

private

our A-block solver

Definition at line 104 of file uzawa_precond.hpp.

◆ _solver_s

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

std::shared_ptr<SolverS> FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_solver_s

private

our S-block solver

Definition at line 106 of file uzawa_precond.hpp.

◆ _uzawa_type

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

UzawaType FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_uzawa_type

private

our Uzawa type

Definition at line 108 of file uzawa_precond.hpp.

Referenced by FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::done_symbolic(), FEAT::Solver::UzawaPrecond< Global::Matrix< MatrixA_, MirrorV_, MirrorV_ >, Global::Matrix< MatrixB_, MirrorV_, MirrorP_ >, Global::Matrix< MatrixD_, MirrorP_, MirrorV_ >, Global::Filter< FilterV_, MirrorV_ >, Global::Filter< FilterP_, MirrorP_ > >::done_symbolic(), FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::init_symbolic(), and FEAT::Solver::UzawaPrecond< Global::Matrix< MatrixA_, MirrorV_, MirrorV_ >, Global::Matrix< MatrixB_, MirrorV_, MirrorP_ >, Global::Matrix< MatrixD_, MirrorP_, MirrorV_ >, Global::Filter< FilterV_, MirrorV_ >, Global::Filter< FilterP_, MirrorP_ > >::init_symbolic().

◆ _vec_tmp_p

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

VectorTypeP FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_vec_tmp_p

private

Definition at line 113 of file uzawa_precond.hpp.

◆ _vec_tmp_v

template<typename MatrixA_ , typename MatrixB_ , typename MatrixD_ , typename FilterV_ , typename FilterP_ >

VectorTypeV FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::_vec_tmp_v

private

a temporary defect vector

Definition at line 112 of file uzawa_precond.hpp.

Referenced by FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::done_symbolic(), and FEAT::Solver::UzawaPrecond< MatrixA_, MatrixB_, MatrixD_, FilterV_, FilterP_ >::init_symbolic().

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

kernel/solver/uzawa_precond.hpp

Public Types

Public Member Functions

Private Attributes

Detailed Description

Member Typedef Documentation

◆ BaseClass

◆ DataType

◆ SolverA

◆ SolverS

◆ VectorType

◆ VectorTypeP

◆ VectorTypeV

Constructor & Destructor Documentation

◆ UzawaPrecond()

◆ ~UzawaPrecond()

Member Function Documentation

◆ apply() [1/2]

◆ apply() [2/2]

◆ done()

◆ done_numeric()

◆ done_symbolic()

◆ init()

◆ init_numeric()

◆ init_symbolic()

◆ name()

Member Data Documentation

◆ _auto_init_s

◆ _filter_p

◆ _filter_v

◆ _matrix_a

◆ _matrix_b

◆ _matrix_d

◆ _solver_a

◆ _solver_s

◆ _uzawa_type

◆ _vec_tmp_p

◆ _vec_tmp_v