To build and solve the linear system for the Poisson problem using the DG P₁ 2-D triangle element. More...

#include <LaplaceDG2DP1.h>

Inheritance diagram for LaplaceDG2DP1:

Public Member Functions
	LaplaceDG2DP1 (Mesh &ms, Vect< real_t > &f, Vect< real_t > &Dbc, Vect< real_t > &Nbc, Vect< real_t > &u)
	Constructor with mesh and vector data.

	~LaplaceDG2DP1 ()
	Destructor.

void	set (real_t sigma, real_t eps)
	Set parameters for the DG method.

void	set (const LocalMatrix< real_t, 2, 2 > &K)
	Set diffusivity matrix.

void	build ()
	Build global matrix and right-hand side.

void	Smooth (Vect< real_t > &u)
	Perform post calculations.

int	run ()
	Build and solve the linear system of equations using an iterative method.

Public Member Functions inherited from DG
	DG (Mesh &ms, size_t degree=1)
	Constructor with mesh and degree of the method.

	~DG ()
	Destructor.

int	setGraph ()
	Set matrix graph.

Public Member Functions inherited from Equation< 3, 3, 2, 2 >
	Equation ()

	Equation (Mesh &mesh)
	Constructor with mesh instance.

	Equation (Mesh &mesh, Vect< real_t > &u)
	Constructor with mesh instance and solution vector.

	Equation (Mesh &mesh, Vect< real_t > &u, real_t &init_time, real_t &final_time, real_t &time_step)
	Constructor with mesh instance, matrix and right-hand side.

	~Equation ()
	Destructor.

void	updateBC (const Element &el, const Vect< real_t > &bc)
	Update Right-Hand side by taking into account essential boundary conditions.

void	DiagBC (DOFSupport dof_type=NODE_DOF, int dof=0)
	Update element matrix to impose bc by diagonalization technique.

void	LocalNodeVector (Vect< real_t > &b)
	Localize element vector from a Vect instance.

void	ElementNodeVector (const Vect< real_t > &b, LocalVect< real_t, NEE_ > &be)
	Localize element vector from a Vect instance.

void	ElementNodeVector (const Vect< real_t > &b, LocalVect< real_t, NEN_ > &be, int dof)
	Localize Element Vector from a Vect instance.

void	SideNodeVector (const Vect< real_t > &b, LocalVect< real_t, NSE_ > &bs)
	Localize side vector from a Vect instance.

void	SideSideVector (const Vect< real_t > &b, vector< real_t > &bs)
	Localize side vector from a Vect instance.

void	ElementNodeVectorSingleDOF (const Vect< real_t > &b, LocalVect< real_t, NEN_ > &be)
	Localize Element Vector from a Vect instance.

void	ElementSideVector (const Vect< real_t > &b, LocalVect< real_t, NSE_ > &be)
	Localize Element Vector from a Vect instance.

void	ElementVector (const Vect< real_t > &b, DOFSupport dof_type=NODE_DOF, int flag=0)
	Localize element vector.

void	SideVector (const Vect< real_t > &b, vector< real_t > &sb)
	Localize side vector.

void	ElementNodeCoordinates ()
	Localize coordinates of element nodes.

void	SideNodeCoordinates ()
	Localize coordinates of side nodes.

void	ElementAssembly (Matrix< real_t > *A)
	Assemble element matrix into global one.

void	ElementAssembly (BMatrix< real_t > &A)
	Assemble element matrix into global one.

void	ElementAssembly (SkSMatrix< real_t > &A)
	Assemble element matrix into global one.

void	ElementAssembly (SkMatrix< real_t > &A)
	Assemble element matrix into global one.

void	ElementAssembly (SpMatrix< real_t > &A)
	Assemble element matrix into global one.

void	ElementAssembly (TrMatrix< real_t > &A)
	Assemble element matrix into global one.

void	ElementAssembly (Vect< real_t > &v)
	Assemble element vector into global one.

void	DGElementAssembly (Matrix< real_t > *A)
	Assemble element matrix into global one for the Discontinuous Galerkin approximation.

void	DGElementAssembly (SkSMatrix< real_t > &A)
	Assemble element matrix into global one for the Discontinuous Galerkin approximation.

void	DGElementAssembly (SkMatrix< real_t > &A)
	Assemble element matrix into global one for the Discontinuous Galerkin approximation.

void	DGElementAssembly (SpMatrix< real_t > &A)
	Assemble element matrix into global one for the Discontinuous Galerkin approximation.

void	DGElementAssembly (TrMatrix< real_t > &A)
	Assemble element matrix into global one for the Discontinuous Galerkin approximation.

void	SideAssembly (Matrix< real_t > *A)
	Assemble side (edge or face) matrix into global one.

void	SideAssembly (SkSMatrix< real_t > &A)
	Assemble side (edge or face) matrix into global one.

void	SideAssembly (SkMatrix< real_t > &A)
	Assemble side (edge or face) matrix into global one.

void	SideAssembly (SpMatrix< real_t > &A)
	Assemble side (edge or face) matrix into global one.

void	SideAssembly (Vect< real_t > &v)
	Assemble side (edge or face) vector into global one.

void	AxbAssembly (const Element &el, const Vect< real_t > &x, Vect< real_t > &b)
	Assemble product of element matrix by element vector into global vector.

void	AxbAssembly (const Side &sd, const Vect< real_t > &x, Vect< real_t > &b)
	Assemble product of side matrix by side vector into global vector.

size_t	getNbNodes () const
	Return number of element nodes.

size_t	getNbEq () const
	Return number of element equations.

real_t	setMaterialProperty (const string &exp, const string &prop)
	Define a material property by an algebraic expression.

Public Member Functions inherited from Equa
	Equa ()
	Default constructor.

virtual	~Equa ()
	Destructor.

void	setMesh (Mesh &m)
	Define mesh and renumber DOFs after removing imposed ones.

Mesh &	getMesh () const
	Return reference to Mesh instance.

LinearSolver &	getLinearSolver ()
	Return reference to linear solver instance.

Matrix< real_t > *	getMatrix () const
	Return pointer to matrix.

void	setSolver (Iteration ls, Preconditioner pc=IDENT_PREC)
	Choose solver for the linear system.

void	setMatrixType (int t)
	Choose type of matrix.

int	solveLinearSystem (Matrix< real_t > *A, Vect< real_t > &b, Vect< real_t > &x)
	Solve the linear system with given matrix and right-hand side.

int	solveLinearSystem (Vect< real_t > &b, Vect< real_t > &x)
	Solve the linear system with given right-hand side.

void	LinearSystemInfo ()
	Print info on linear system solver.

Detailed Description

To build and solve the linear system for the Poisson problem using the DG P₁ 2-D triangle element.

This class build the linear system of equations for a standard elliptic equation using the Discontinuous Galerkin P₁ finite element method.

Author: Rachid Touzani

Copyright: GNU Lesser Public License

Constructor & Destructor Documentation

◆ LaplaceDG2DP1()

LaplaceDG2DP1	(	Mesh &	ms,
		Vect< real_t > &	f,
		Vect< real_t > &	Dbc,
		Vect< real_t > &	Nbc,
		Vect< real_t > &	u
	)

Constructor with mesh and vector data.

Parameters

[in]	ms	Mesh instance
[in]	f	Vector containing the right-hand side of the elliptic equation at triangle vertices
[in]	Dbc	Vector containing prescribed values of the solution (Dirichlet boundary condition) at nodes having a positive code
[in]	Nbc	Vector containing prescribed values of the flux (Neumann boundary condition) at each side having a positive code
[in]	u	Vector where the solution is stored once the linear system is solved

Member Function Documentation

◆ build()

void build ( )

Build global matrix and right-hand side.

The problem matrix and right-hand side are the ones used in the constructor. They are updated in this member function.

◆ run()

int run ( )

Build and solve the linear system of equations using an iterative method.

The matrix is preconditioned by the diagonal ILU method. The linear system is solved either by the Conjugate Gradient method if the matrix is symmetric positive definite (eps=-1) or the GMRES method if not. The solution is stored in the vector u given in the constructor.

Returns: Number of performed iterations. Note that the maximal number is 1000 and the tolerance is 1.e-8

◆ set() [1/2]

void set ( const LocalMatrix< real_t, 2, 2 > & K )

Set diffusivity matrix.

This function provides the diffusivity matrix as instance of class LocalMatrix. The default diffusivity matrix is the identity matrix

Parameters

[in] K Diffusivity matrix

◆ set() [2/2]

void set	(	real_t	sigma,
		real_t	eps
	)

Set parameters for the DG method.

Parameters

[in]	sigma	Penalty parameters to enforce continuity at nodes (Must be positive) [Default: `100`]
[in]	eps	Epsilon value of the DG method to choose among the values: 0 Incomplete Interior Penalty Galerkin method (IIPG) -1 Symmetric Interior Penalty Galerkin method (SIPG) 1 Non symmetric interior penalty Galerkin method (NIPG) For a user not familiar with the method, please choose the value of `eps=-1` and `sigma>100` which leads to a symmetric positive definite matrix [Default: `-1`]

◆ Smooth()

void Smooth ( Vect< real_t > & u )

Perform post calculations.

This function gives an averaged solution given at mesh nodes (triangle vertices) by a standard L₂-projection method.

Parameters

[in] u Solution at nodes

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ LaplaceDG2DP1()

Member Function Documentation

◆ build()

◆ run()

◆ set() [1/2]

◆ set() [2/2]

◆ Smooth()