Local solver for Poisson's equation on uniform hypergraph. More...

#include <diffusion_uniform_ldgh.hxx>

Collaboration diagram for LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >:

Classes
struct	data_type
	Define type of (hyperedge related) data that is stored in HyDataContainer. More...

struct	error_def
	Define how errors are evaluated. More...

struct	node_element
	Define type of node elements, especially with respect to nodal shape functions. More...

Public Types
typedef lSol_float_t	constructor_value_type
	Class is constructed using a single double indicating the penalty parameter. More...

Public Member Functions
	DiffusionUniform (const constructor_value_type &tau=1.)
	Constructor for local solver. More...

template<typename SmallMatInT , typename SmallMatOutT >
SmallMatOutT &	trace_to_flux (const SmallMatInT &lambda_values_in, SmallMatOutT &lambda_values_out, const lSol_float_t UNUSED(time)=0.) const
	Evaluate local contribution to matrix–vector multiplication. More...

template<typename SmallMatInT , typename SmallMatOutT >
SmallMatOutT &	residual_flux (const SmallMatInT &lambda_values_in, SmallMatOutT &lambda_values_out, const lSol_float_t UNUSED(time)=0.) const
	Evaluate local contribution to matrix–vector residual. More...

template<typename SmallMatT >
std::array< lSol_float_t, 1U >	errors (const SmallMatT &UNUSED(lambda_values), const lSol_float_t UNUSED(time)=0.) const
	Evaluate local squared L2 error. More...

template<typename abscissa_float_t , std::size_t abscissas_sizeT, class input_array_t >
std::array< std::array< lSol_float_t, Hypercube< hyEdge_dimT >::pow(abscissas_sizeT)>, system_dim >	bulk_values (const std::array< abscissa_float_t, abscissas_sizeT > &abscissas, const input_array_t &lambda_values, const lSol_float_t UNUSED(time)=0.) const
	Evaluate local reconstruction at tensorial products of abscissas. More...

Static Public Member Functions
static constexpr unsigned int	hyEdge_dim ()
	Dimension of hyper edge type that this object solves on. More...

static constexpr unsigned int	n_glob_dofs_per_node ()
	Evaluate amount of global degrees of freedom per hypernode. More...

static constexpr unsigned int	system_dimension ()
	Dimension of of the solution evaluated with respect to a hyperedge. More...

static constexpr unsigned int	node_system_dimension ()
	Dimension of of the solution evaluated with respect to a hypernode. More...

Private Types
typedef TPP::Quadrature::Tensorial< TPP::Quadrature::GaussLegendre< quad_deg >, TPP::ShapeFunction< TPP::ShapeType::Tensorial< TPP::ShapeType::Legendre< poly_deg >, hyEdge_dimT > >, lSol_float_t >	integrator
	An integrator helps to easily evaluate integrals (e.g. via quadrature). More...

Private Member Functions
template<typename SmallMatT >
SmallVec< n_loc_dofs_, lSol_float_t >	assemble_rhs (const SmallMatT &lambda_values) const
	Assemble local right hand for the local solver. More...

template<typename SmallMatT >
std::array< lSol_float_t, n_loc_dofs_ >	solve_local_problem (const SmallMatT &lambda_values) const
	Solve local problem. More...

std::array< std::array< lSol_float_t, n_shape_bdr_ >, 2 *hyEdge_dimT >	primal_at_boundary (const std::array< lSol_float_t, n_loc_dofs_ > &coeffs) const
	Evaluate primal variable at boundary. More...

std::array< std::array< lSol_float_t, n_shape_bdr_ >, 2 *hyEdge_dimT >	dual_at_boundary (const std::array< lSol_float_t,(hyEdge_dimT+1) *n_shape_fct_ > &coeffs) const
	Evaluate dual variable at boundary. More...

Static Private Member Functions
static SmallSquareMat< n_loc_dofs_, lSol_float_t >	assemble_loc_matrix (const lSol_float_t tau)
	Assemble local matrix for the local solver. More...

Private Attributes
const lSol_float_t	tau_
	(Globally constant) penalty parameter for HDG scheme. More...

const SmallSquareMat< n_loc_dofs_, lSol_float_t >	loc_mat_
	Local matrix for the local solver. More...

Static Private Attributes
static constexpr unsigned int	n_shape_fct_ = n_glob_dofs_per_node() * (poly_deg + 1)
	Number of local shape functions (with respect to all spatial dimensions). More...

static constexpr unsigned int	n_shape_bdr_ = n_glob_dofs_per_node()
	Number oflocal shape functions (with respect to a face / hypernode). More...

static constexpr unsigned int	n_loc_dofs_ = (hyEdge_dimT + 1) * n_shape_fct_
	Number of (local) degrees of freedom per hyperedge. More...

static constexpr unsigned int	system_dim = system_dimension()
	Dimension of of the solution evaluated with respect to a hypernode. More...

static constexpr unsigned int	node_system_dim = node_system_dimension()
	Dimension of of the solution evaluated with respect to a hypernode. More...

Detailed Description

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>
class LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >

Local solver for Poisson's equation on uniform hypergraph.

This class contains the local solver for Poisson's equation, i.e.,

\[ - \Delta u = 0 \quad \text{ in } \Omega, \qquad u = u_\text D \quad \text{ on } \partial \Omega \]

in a spatial domain \(\Omega \subset \mathbb R^d\). Here, \(d\) is the spatial dimension space_dim, \(\Omega\) is a regular graph (hyEdge_dimT = 1), or hypergraph whose hyperedges are surfaces (hyEdge_dimT = 2), or volumes (hyEdge_dimT = 3), or higher dimensional analogues. For this class, all hyperedges are supposed to be uniform (i.e. equal to the unit hypercube). Thus, no geometrical information is needed by this class.

Template Parameters

hyEdge_dimT	Dimension of a hyperedge, i.e., 1 is for PDEs defined on graphs, 2 is for PDEs defined on surfaces, and 3 is for PDEs defined on volumes.
poly_deg	The polynomial degree of test and trial functions.
quad_deg	The order of the quadrature rule.
lSol_float_t	The floating point type calculations are executed in. Defaults to double.

Authors: Guido Kanschat, Heidelberg University, 2019–2020.; Andreas Rupp, Heidelberg University, 2019–2020.

Member Typedef Documentation

◆ constructor_value_type

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

typedef lSol_float_t LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::constructor_value_type

Class is constructed using a single double indicating the penalty parameter.

◆ integrator

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

typedef TPP::Quadrature::Tensorial< TPP::Quadrature::GaussLegendre<quad_deg>, TPP::ShapeFunction<TPP::ShapeType::Tensorial<TPP::ShapeType::Legendre<poly_deg>, hyEdge_dimT> >, lSol_float_t> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::integrator

private

An integrator helps to easily evaluate integrals (e.g. via quadrature).

Constructor & Destructor Documentation

◆ DiffusionUniform()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::DiffusionUniform ( const constructor_value_type & tau = 1. )

inline

Constructor for local solver.

Parameters

tau	Penalty parameter of HDG scheme.

Member Function Documentation

◆ assemble_loc_matrix()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

static SmallSquareMat<n_loc_dofs_, lSol_float_t> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::assemble_loc_matrix ( const lSol_float_t tau )

staticprivate

Assemble local matrix for the local solver.

The local solver neither depends on the geometry, nor on global functions. Thus, its local matrix is the same for all hyperedges and can be assembled once in the constructor. This is done in this function.

The function is static, since it is used in the constructor's initializer list.

Parameters

tau	Penalty parameter for HDG.

Return values

loc_mat Matrix of the local solver.

◆ assemble_rhs()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename SmallMatT >

SmallVec<n_loc_dofs_, lSol_float_t> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::assemble_rhs ( const SmallMatT & lambda_values ) const

inlineprivate

Assemble local right hand for the local solver.

The right hand side needs the values of the global degrees of freedom. Thus, it needs to be constructed individually for every hyperedge.

Template Parameters

SmallMatT The data type of the lambda values.

Parameters

lambda_values Global degrees of freedom associated to the hyperedge.

Return values

loc_rhs Local right hand side of the locasl solver.

◆ bulk_values()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename abscissa_float_t , std::size_t abscissas_sizeT, class input_array_t >

std::array<std::array<lSol_float_t, Hypercube<hyEdge_dimT>::pow(abscissas_sizeT)>, system_dim> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::bulk_values	(	const std::array< abscissa_float_t, abscissas_sizeT > &	abscissas,
		const input_array_t &	lambda_values,
		const lSol_float_t	UNUSEDtime = `0.`
	)		const

Evaluate local reconstruction at tensorial products of abscissas.

Template Parameters

absc_float_t	Floating type for the abscissa values.
abscissas_sizeT	Size of the array of array of abscissas.
input_array_t	Type of input array.

Parameters

abscissas	Abscissas of the supporting points.
lambda_values	The values of the skeletal variable's coefficients.
time	Time — this parameter is redundant for this local solver.

Return values

function_values Function values at quadrature points.

◆ dual_at_boundary()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

std::array<std::array<lSol_float_t, n_shape_bdr_>, 2 * hyEdge_dimT> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::dual_at_boundary ( const std::array< lSol_float_t,(hyEdge_dimT+1) *n_shape_fct_ > & coeffs ) const

inlineprivate

Evaluate dual variable at boundary.

Function to evaluate dual variable of the solution. This function is needed to calculate the local numerical fluxes.

Parameters

coeffs Coefficients of the local solution.

Return values

bdr_coeffs Coefficients of respective (dim-1) dimensional function at boundaries.

◆ errors()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename SmallMatT >

std::array<lSol_float_t, 1U> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::errors	(	const SmallMatT &	UNUSEDlambda_values,
		const lSol_float_t	UNUSEDtime = `0.`
	)		const

inline

Evaluate local squared L2 error.

Template Parameters

SmallMatT The typename of lambda_values.

Parameters

lambda_values	The values of the skeletal variable's coefficients.
time	Time — this parameter is redundant for this local solver.

Return values

vec_b Local part of vector b.

◆ hyEdge_dim()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

static constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::hyEdge_dim ( )

inlinestaticconstexpr

Dimension of hyper edge type that this object solves on.

◆ n_glob_dofs_per_node()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

static constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::n_glob_dofs_per_node ( )

inlinestaticconstexpr

Evaluate amount of global degrees of freedom per hypernode.

This number must be equal to HyperNodeFactory::n_dofs_per_node() of the HyperNodeFactory cooperating with this object.

Return values

n_dofs Number of global degrees of freedom per hypernode.

◆ node_system_dimension()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

static constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::node_system_dimension ( )

inlinestaticconstexpr

Dimension of of the solution evaluated with respect to a hypernode.

◆ primal_at_boundary()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

std::array<std::array<lSol_float_t, n_shape_bdr_>, 2 * hyEdge_dimT> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::primal_at_boundary ( const std::array< lSol_float_t, n_loc_dofs_ > & coeffs ) const

inlineprivate

Evaluate primal variable at boundary.

Function to evaluate primal variable of the solution. This function is needed to calculate the local numerical fluxes.

Parameters

coeffs Coefficients of the local solution.

Return values

bdr_coeffs Coefficients of respective (dim-1) dimensional function at boundaries.

◆ residual_flux()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename SmallMatInT , typename SmallMatOutT >

SmallMatOutT& LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::residual_flux	(	const SmallMatInT &	lambda_values_in,
		SmallMatOutT &	lambda_values_out,
		const lSol_float_t	UNUSEDtime = `0.`
	)		const

inline

Evaluate local contribution to matrix–vector residual.

Template Parameters

SmallMatInT	Data type of `lambda_values_in`.
SmallMatOutT	Data type of `lambda_values_out`

Parameters

lambda_values_in	Local part of vector x.
lambda_values_out	Local part that will be added to A * x.
time	Time — this parameter is redundant for this local solver.

Return values

vecAx Local part of vector A * x.

◆ solve_local_problem()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename SmallMatT >

std::array<lSol_float_t, n_loc_dofs_> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::solve_local_problem ( const SmallMatT & lambda_values ) const

inlineprivate

Solve local problem.

Template Parameters

SmallMatT The data type of the lambda values.

Parameters

lambda_values Global degrees of freedom associated to the hyperedge.

Return values

loc_sol Solution of the local problem.

◆ system_dimension()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

static constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::system_dimension ( )

inlinestaticconstexpr

Dimension of of the solution evaluated with respect to a hyperedge.

◆ trace_to_flux()

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

template<typename SmallMatInT , typename SmallMatOutT >

SmallMatOutT& LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::trace_to_flux	(	const SmallMatInT &	lambda_values_in,
		SmallMatOutT &	lambda_values_out,
		const lSol_float_t	UNUSEDtime = `0.`
	)		const

inline

Evaluate local contribution to matrix–vector multiplication.

Execute matrix–vector multiplication y = A * x, where x represents the vector containing the skeletal variable (adjacent to one hyperedge), and A is the condensed matrix arising from the HDG discretization. This function does this multiplication (locally) for one hyperedge. The hyperedge is no parameter, since all hyperedges are assumed to have the same properties.

Template Parameters

SmallMatInT	Data type of `lambda_values_in`.
SmallMatOutT	Data type of `lambda_values_out`

Parameters

lambda_values_in	Local part of vector x.
lambda_values_out	Local part that will be added to A * x.
time	Time — this parameter is redundant for this local solver.

Return values

vecAx Local part of vector A * x.

Member Data Documentation

◆ loc_mat_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

const SmallSquareMat<n_loc_dofs_, lSol_float_t> LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::loc_mat_

private

Local matrix for the local solver.

◆ n_loc_dofs_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::n_loc_dofs_ = (hyEdge_dimT + 1) * n_shape_fct_

staticconstexprprivate

Number of (local) degrees of freedom per hyperedge.

◆ n_shape_bdr_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::n_shape_bdr_ = n_glob_dofs_per_node()

staticconstexprprivate

Number oflocal shape functions (with respect to a face / hypernode).

◆ n_shape_fct_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::n_shape_fct_ = n_glob_dofs_per_node() * (poly_deg + 1)

staticconstexprprivate

Number of local shape functions (with respect to all spatial dimensions).

◆ node_system_dim

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::node_system_dim = node_system_dimension()

staticconstexprprivate

Dimension of of the solution evaluated with respect to a hypernode.

This allows to the use of this quantity as template parameter in member functions.

◆ system_dim

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

constexpr unsigned int LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::system_dim = system_dimension()

staticconstexprprivate

Dimension of of the solution evaluated with respect to a hypernode.

This allows to the use of this quantity as template parameter in member functions.

◆ tau_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>

const lSol_float_t LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >::tau_

private

(Globally constant) penalty parameter for HDG scheme.

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

/home/runner/work/HyperHDG/HyperHDG/include/HyperHDG/local_solver/diffusion_uniform_ldgh.hxx

Classes

Public Types

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double> class LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >

Member Typedef Documentation

◆ constructor_value_type

◆ integrator

Constructor & Destructor Documentation

◆ DiffusionUniform()

Member Function Documentation

◆ assemble_loc_matrix()

◆ assemble_rhs()

◆ bulk_values()

◆ dual_at_boundary()

◆ errors()

◆ hyEdge_dim()

◆ n_glob_dofs_per_node()

◆ node_system_dimension()

◆ primal_at_boundary()

◆ residual_flux()

◆ solve_local_problem()

◆ system_dimension()

◆ trace_to_flux()

Member Data Documentation

◆ loc_mat_

◆ n_loc_dofs_

◆ n_shape_bdr_

◆ n_shape_fct_

◆ node_system_dim

◆ system_dim

◆ tau_

template<unsigned int hyEdge_dimT, unsigned int poly_deg, unsigned int quad_deg, typename lSol_float_t = double>
class LocalSolver::DiffusionUniform< hyEdge_dimT, poly_deg, quad_deg, lSol_float_t >