Example solution of a div-curl system on a hexahedral mesh using div-conforming (face) elements. More...

#include "TrilinosCouplings_config.h"
#include "TrilinosCouplings_Pamgen_Utils.hpp"
#include "Intrepid_FunctionSpaceTools.hpp"
#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_CellTools.hpp"
#include "Intrepid_ArrayTools.hpp"
#include "Intrepid_HCURL_HEX_I1_FEM.hpp"
#include "Intrepid_HGRAD_HEX_C1_FEM.hpp"
#include "Intrepid_HDIV_HEX_I1_FEM.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_DefaultCubatureFactory.hpp"
#include "Intrepid_Utils.hpp"
#include "Epetra_Time.h"
#include "Epetra_Map.h"
#include "Epetra_SerialComm.h"
#include "Epetra_FECrsMatrix.h"
#include "Epetra_FEVector.h"
#include "Epetra_Vector.h"
#include "Epetra_LinearProblem.h"
#include "Epetra_Import.h"
#include "Epetra_Export.h"
#include "Teuchos_oblackholestream.hpp"
#include "Teuchos_RCP.hpp"
#include "Teuchos_BLAS.hpp"
#include "Teuchos_GlobalMPISession.hpp"
#include "Teuchos_ParameterList.hpp"
#include "Teuchos_XMLParameterListHelpers.hpp"
#include "Shards_CellTopology.hpp"
#include "EpetraExt_RowMatrixOut.h"
#include "EpetraExt_MultiVectorOut.h"
#include "EpetraExt_VectorOut.h"
#include "EpetraExt_MatrixMatrix.h"
#include "create_inline_mesh.h"
#include "pamgen_im_exodusII_l.h"
#include "pamgen_im_ne_nemesisI_l.h"
#include "pamgen_extras.h"
#include "AztecOO.h"
#include "ml_epetra_utils.h"
#include "ml_RefMaxwell_11_Operator.h"
#include "ml_FaceMatrixFreePreconditioner.h"
#include "ml_RefMaxwell.h"

Include dependency graph for example_DivLSFEM.cpp:

Classes
struct	fecomp

Macros
#define	ABS(x) ((x)>0?(x):-(x))

Typedefs
typedef Intrepid::FunctionSpaceTools	IntrepidFSTools

typedef Intrepid::RealSpaceTools < double >	IntrepidRSTools

typedef Intrepid::CellTools < double >	IntrepidCTools

Functions
int	Multiply_Abs (const Epetra_CrsMatrix &A, const Epetra_Vector &x, Epetra_Vector &y)
	Multiplies abs(A)x = y, where all non-zero entries of A are replaced with their absolute values value. More...

void	TestMultiLevelPreconditioner_DivLSFEM (char ProblemType[], Teuchos::ParameterList &MLList, Epetra_CrsMatrix &GradDiv, Epetra_CrsMatrix &D0clean, Epetra_CrsMatrix &D1clean, Epetra_CrsMatrix &FaceNode, Epetra_CrsMatrix &M1, Epetra_CrsMatrix &M1inv, Epetra_CrsMatrix &M2, Epetra_MultiVector &xh, Epetra_MultiVector &b, double &TotalErrorResidual, double &TotalErrorExactSol)
	ML Preconditioner. More...

int	evalu (double &uExact0, double &uExact1, double &uExact2, double &x, double &y, double &z)
	Exact solution evaluation. More...

double	evalDivu (double &x, double &y, double &z)
	Divergence of exact solution. More...

int	evalCurlu (double &curlu0, double &curlu1, double &curlu2, double &x, double &y, double &z, double &mu)
	Curl of exact solution. More...

int	evalCurlCurlu (double &curlCurlu0, double &curlCurlu1, double &curlCurlu2, double &x, double &y, double &z, double &mu)
	Curl of curl of exact solution. More...

int	main (int argc, char *argv[])

int	Multiply_Ones (const Epetra_CrsMatrix &A, const Epetra_Vector &x, Epetra_Vector &y)
	Multiplies Ax = y, where all non-zero entries of A are replaced with the value 1.0. More...

void	solution_test (string msg, const Epetra_Operator &A, const Epetra_MultiVector &lhs, const Epetra_MultiVector &rhs, const Epetra_MultiVector &xexact, Epetra_Time &Time, double &TotalErrorExactSol, double &TotalErrorResidual)
	Compute ML solution residual. More...

Detailed Description

Example solution of a div-curl system on a hexahedral mesh using div-conforming (face) elements.

    This example uses Pamgen to generate a hexahedral mesh, Intrepid to
    build mass and stiffness matrices, and ML to solve.

       System

                   div v + \phi = f  in Omega
                   v+ A grad \phi = 0  in Omega
                   Dirichlet BC: \phi given  on Gamma

                   Omega is the box [0,1]^3

        where f is derived from a prescribed solution

        \phi(x,y,z)=-sin^2(\pi x)*sin^2(\pi y)*sin^2(\pi z)

        or other, discontinuous solution, aka "patch test"

Remarks

Usage

 ./TrilinosCouplings_examples_scaling_Example_DivLSFEM.exe  inputfile.xml

    inputfile.xml (optional)  -  xml input file containing Pamgen mesh description
                                 and material parameters for each Pamgen block,
                                 if not present code attempts to read DivLSFEMin.xml.

  This example uses the following Trilinos packages:

Pamgen to generate a Hexahedral mesh.
Intrepid to build the discretization matrices and right-hand side.
Epetra to handle the global matrix and vector.
ML to solve the linear system.

 For more details on the formulation see

P. Bochev, K. Peterson, C. Siefert, "Analysis and Computation of Compatible Least-Squares Methods for Div-Curl Equations", SIAM J. Numerical Analysis, vol 49, pp 159-191, 2011.

        Div-Curl System:

                   curl u = g  in Omega
                    div u = h  in Omega
                      u.n = 0  on Gamma

        Corresponding discrete linear system for face element coeficients (x):

                  (Kd + Md*Dc*McInv*Dc'*Md)x = b

                  Kd    - Hdiv stiffness matrix
                  Md    - Hdiv mass matrix
                  Dc    - Edge to Face incidence matrix
                  McInv - Hgrad mass matrix inverse
                  b     - right hand side vector

Author: Created by P. Bochev, D. Ridzal, K. Peterson, D. Hensinger, C. Siefert.

Remarks

Usage

 ./TrilinosCouplings_examples_scaling_Example_DivLSFEM.exe  inputfile.xml

    inputfile.xml (optional)  -  xml input file containing Pamgen mesh description
                                 and material parameters for each Pamgen block,
                                 if not present code attempts to read DivLSFEMin.xml.

Function Documentation

int evalCurlCurlu	(	double &	curlCurlu0,
		double &	curlCurlu1,
		double &	curlCurlu2,
		double &	x,
		double &	y,
		double &	z,
		double &	mu
	)

Curl of curl of exact solution.

Parameters

curlCurlu0	[out] first component of curl curl of exact solution
curlCurlu1	[out] second component of curl curl of exact solution
curlCurlu2	[out] third component of curl curl of exact solution
x	[in] x coordinate
y	[in] y coordinate
z	[in] z coordinate
mu	[in] material parameter

int evalCurlu	(	double &	curlu0,
		double &	curlu1,
		double &	curlu2,
		double &	x,
		double &	y,
		double &	z,
		double &	mu
	)

Curl of exact solution.

Parameters

curlu0	[out] first component of curl of exact solution
curlu1	[out] second component of curl of exact solution
curlu2	[out] third component of curl of exact solution
x	[in] x coordinate
y	[in] y coordinate
z	[in] z coordinate
mu	[in] material parameter

double evalDivu	(	double &	x,
		double &	y,
		double &	z
	)

Divergence of exact solution.

Parameters

x	[in] x coordinate
y	[in] y coordinate
z	[in] z coordinate

Returns: Value of the divergence of exact solution at (x,y,z)

int evalu	(	double &	uExact0,
		double &	uExact1,
		double &	uExact2,
		double &	x,
		double &	y,
		double &	z
	)

Exact solution evaluation.

Parameters

uExact0	[out] first component of exact solution at (x,y,z)
uExact1	[out] second component of exact solution at (x,y,z)
uExact2	[out] third component of exact solution at (x,y,z)
x	[in] x coordinate
y	[in] y coordinate
z	[in] z coordinate

int Multiply_Abs	(	const Epetra_CrsMatrix &	A,
		const Epetra_Vector &	x,
		Epetra_Vector &	y
	)

Multiplies abs(A)x = y, where all non-zero entries of A are replaced with their absolute values value.

Parameters

A	[in] matrix
x	[in] vector
y	[out] vector

int Multiply_Ones	(	const Epetra_CrsMatrix &	A,
		const Epetra_Vector &	x,
		Epetra_Vector &	y
	)

Multiplies Ax = y, where all non-zero entries of A are replaced with the value 1.0.

Parameters

A	[in] matrix
x	[in] vector
y	[in] vector

void solution_test	(	string	msg,
		const Epetra_Operator &	A,
		const Epetra_MultiVector &	lhs,
		const Epetra_MultiVector &	rhs,
		const Epetra_MultiVector &	xexact,
		Epetra_Time &	Time,
		double &	TotalErrorExactSol,
		double &	TotalErrorResidual
	)

Compute ML solution residual.

Parameters

A	[in] discrete operator
lhs	[in] solution vector
rhs	[in] right hand side vector
Time	[in] elapsed time for output
TotalErrorResidual	[out] error residual
TotalErrorExactSol	[out] error in xh (not an appropriate measure for H(div) basis functions)

void TestMultiLevelPreconditioner_DivLSFEM	(	char	ProblemType[],
		Teuchos::ParameterList &	MLList,
		Epetra_CrsMatrix &	GradDiv,
		Epetra_CrsMatrix &	D0clean,
		Epetra_CrsMatrix &	D1clean,
		Epetra_CrsMatrix &	FaceNode,
		Epetra_CrsMatrix &	M1,
		Epetra_CrsMatrix &	M1inv,
		Epetra_CrsMatrix &	M2,
		Epetra_MultiVector &	xh,
		Epetra_MultiVector &	b,
		double &	TotalErrorResidual,
		double &	TotalErrorExactSol
	)

ML Preconditioner.

Parameters

ProblemType	[in] problem type
MLList	[in] ML parameter list
GradDiv	[in] H(curl) stiffness matrix
D0clean	[in] Edge to node incidence matrix
D1clean	[in] Face to edge incidence matrix
FaceNode	[in] Face to node incidence matrix
M1inv	[in] H(curl) mass matrix inverse
M2	[in] H(div) mass matrix
xh	[out] solution vector
b	[in] right-hand-side vector
TotalErrorResidual	[out] error residual
TotalErrorExactSol	[out] error in xh

References Multiply_Ones(), and solution_test().

Classes

Macros

Typedefs

Functions

Detailed Description

Function Documentation