HybridCore

Classes providing cell and edge quadrature rules, and values of basis functions at the nodes. More...

Classes
class	HArDCore2D::ElementQuad
class	HArDCore2D::UVector
class	HArDCore2D::HybridCore

Typedefs
typedef Family< MonomialScalarBasisCell >	HArDCore2D::HybridCore::PolyCellBasisType
	type for cell basis More...
typedef Family< MonomialScalarBasisEdge >	HArDCore2D::HybridCore::PolyEdgeBasisType
	type for edge basis More...

Functions
	HArDCore2D::ElementQuad::ElementQuad (const HybridCore &hho, const size_t iT, const size_t doeT, const size_t doeF)
	Class constructor: loads the quadrature rules and values of basis functions/gradients at these points. More...
const QuadratureRule &	HArDCore2D::ElementQuad::get_quadT () const
	Returns quadrature rules in cell. More...
const QuadratureRule &	HArDCore2D::ElementQuad::get_quadF (size_t ilF) const
	Returns quadrature rules on edge with local number ilF. More...
const boost::multi_array< double, 2 > &	HArDCore2D::ElementQuad::get_phiT_quadT () const
	Returns values of cell basis functions at cell quadrature nodes. More...
const boost::multi_array< VectorRd, 2 > &	HArDCore2D::ElementQuad::get_dphiT_quadT () const
	Returns values of gradients of cell basis functions at cell quadrature nodes. More...
const boost::multi_array< double, 2 > &	HArDCore2D::ElementQuad::get_phiT_quadF (size_t ilF) const
	Returns values of cell basis functions at edge quadrature nodes, for edge with local number ilF. More...
const boost::multi_array< double, 2 >	HArDCore2D::ElementQuad::get_phiF_quadF (size_t ilF) const
	Returns values of edge basis functions at edge quadrature nodes, for edge with local number ilF. More...
const boost::multi_array< VectorRd, 2 > &	HArDCore2D::ElementQuad::get_dphiT_quadF (size_t ilF) const
	Returns values of gradients of cell basis functions at edge quadrature nodes, for edge with local number ilF. More...
boost::multi_array< VectorRd, 2 >	HArDCore2D::ElementQuad::get_vec_phiT_quadT (size_t degree) const
	Builds on the fly the values of vector cell basis functions at cell quadrature nodes. The vector basis is obtained by tensorization of the scalar one: \((\phi_1,0), (\phi_2,0), ..., (\phi_N,0), (0,\phi_1), (0,\phi_2) ... (0,\phi_N)\). More...
boost::multi_array< VectorRd, 2 >	HArDCore2D::ElementQuad::get_vec_phiT_quadF (size_t ilF, size_t degree) const
	Builds on the fly the values of vector cell basis functions at edge quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT. More...
boost::multi_array< VectorRd, 2 >	HArDCore2D::ElementQuad::get_vec_phiF_quadF (size_t ilF, size_t degree) const
	Builds on the fly the values of vector edge basis functions at edge quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT. More...
template<typename GeometricSupport >
size_t const	HArDCore2D::DimPoly (int m)
template<>
const size_t	HArDCore2D::DimPoly< Cell > (const int m)
	Compute the size of the basis of 2-variate polynomials up to degree m. More...
template<>
const size_t	HArDCore2D::DimPoly< Edge > (const int m)
	Compute the size of the basis of 1-variate polynomials up to degree m. More...
	HArDCore2D::UVector::UVector (const Eigen::VectorXd &values, const Mesh &mesh, const int cell_deg, const size_t edge_deg)
Eigen::VectorXd &	HArDCore2D::UVector::asVectorXd () const
	Return the values as an Eigen vector. More...
const int	HArDCore2D::UVector::get_cell_deg () const
	Return the cell degree. More...
const size_t	HArDCore2D::UVector::get_edge_deg () const
	Return the edge degree. More...
const size_t	HArDCore2D::UVector::n_cell_dofs () const
	Number of dofs in each cell. More...
const size_t	HArDCore2D::UVector::n_edge_dofs () const
	Number of dofs on each edge. More...
const size_t	HArDCore2D::UVector::n_total_cell_dofs () const
	Total number of cell dofs (in the vector, this is where the edge dofs start) More...
Eigen::VectorXd	HArDCore2D::UVector::restr (size_t iT) const
	Extract the restriction of the unknowns corresponding to cell iT and its edges. More...
UVector	HArDCore2D::UVector::operator+ (const UVector &b)
	Overloads the addition: adds the coefficients. More...
UVector	HArDCore2D::UVector::operator- (const UVector &b)
	Overloads the subtraction: subtracts the coefficients. More...
void	HArDCore2D::UVector::operator+= (const UVector &b)
	Overloads the increment: adds the coefficients. More...
void	HArDCore2D::UVector::operator/= (const double &r)
	Overloads the increment: adds the coefficients. More...
double	HArDCore2D::UVector::operator() (size_t index) const
	Overloads the (): returns the corresponding coefficient. More...
	HArDCore2D::HybridCore::HybridCore (const Mesh *mesh_ptr, const int cell_deg, const size_t edge_deg, const bool use_threads=true, std::ostream &output=std::cout, const bool ortho=true)
	Class constructor: initialises the data structure with the given mesh, and desired polynomial degrees of the basis functions. More...
const Mesh *	HArDCore2D::HybridCore::get_mesh () const
	Returns a pointer to the mesh. More...
const int	HArDCore2D::HybridCore::CellDegree () const
	Return the degree of cell polynomials. More...
const int	HArDCore2D::HybridCore::CellDegreePos () const
const size_t	HArDCore2D::HybridCore::EdgeDegree () const
	Return the degree of edge polynomials. More...
const PolyCellBasisType &	HArDCore2D::HybridCore::CellBasis (size_t iT) const
	Return cell basis for element with global index iT. More...
const PolyEdgeBasisType &	HArDCore2D::HybridCore::EdgeBasis (size_t iE) const
	Return edge basis for edge with global index iE. More...
double	HArDCore2D::HybridCore::L2norm (const UVector &Xh) const
	Compute L2 norm of a discrete function (using cell values) More...
double	HArDCore2D::HybridCore::H1norm (const UVector &Xh) const
	Compute discrete H1 norm of a discrete function. More...
template<typename ContinuousFunction >
UVector	HArDCore2D::HybridCore::interpolate (const ContinuousFunction &f, const int deg_cell, const size_t deg_edge, size_t doe) const
	Compute the interpolant in the discrete space of a continuous function. More...
Eigen::VectorXd	HArDCore2D::HybridCore::compute_weights (size_t iT) const
	Computes the weights to get cell values from edge values when l=-1. More...
double	HArDCore2D::HybridCore::evaluate_in_cell (const UVector Xh, size_t iT, VectorRd x) const
	Evaluates a discrete function in the cell iT at point x. More...
double	HArDCore2D::HybridCore::evaluate_in_edge (const UVector Xh, size_t iE, VectorRd x) const
	Evaluates a discrete function on the edge iE at point x. More...
Eigen::VectorXd	HArDCore2D::HybridCore::VertexValues (const UVector Xh, const std::string from_dofs)
	From a hybrid function, computes a vector of values at the vertices of the mesh. More...

Detailed Description

Classes providing cell and edge quadrature rules, and values of basis functions at the nodes.

Classes providing tools to implement schemes having polynomial unknowns in the cells and on the edges.

Typedef Documentation

PolyCellBasisType

typedef Family<MonomialScalarBasisCell> HArDCore2D::HybridCore::PolyCellBasisType

type for cell basis

PolyEdgeBasisType

typedef Family<MonomialScalarBasisEdge> HArDCore2D::HybridCore::PolyEdgeBasisType

type for edge basis

Function Documentation

asVectorXd()

Eigen::VectorXd& HArDCore2D::UVector::asVectorXd ( ) const

inline

Return the values as an Eigen vector.

CellBasis()

const PolyCellBasisType& HArDCore2D::HybridCore::CellBasis ( size_t  iT ) const

inline

Return cell basis for element with global index iT.

CellDegree()

const int HArDCore2D::HybridCore::CellDegree ( ) const

inline

Return the degree of cell polynomials.

CellDegreePos()

const int HArDCore2D::HybridCore::CellDegreePos ( ) const

inline

compute_weights()

Eigen::VectorXd HybridCore::compute_weights

(

size_t

iT

)

const

Computes the weights to get cell values from edge values when l=-1.

DimPoly()

template<typename GeometricSupport >

size_t const HArDCore2D::DimPoly ( int  m )

inline

DimPoly< Cell >()

template<>

const size_t HArDCore2D::DimPoly< Cell > ( const int  m )

inline

Compute the size of the basis of 2-variate polynomials up to degree m.

Parameters

m	Polynomial degree

DimPoly< Edge >()

template<>

const size_t HArDCore2D::DimPoly< Edge > ( const int  m )

inline

Compute the size of the basis of 1-variate polynomials up to degree m.

Parameters

m	Polynomial degree

EdgeBasis()

const PolyEdgeBasisType& HArDCore2D::HybridCore::EdgeBasis ( size_t  iE ) const

inline

Return edge basis for edge with global index iE.

EdgeDegree()

const size_t HArDCore2D::HybridCore::EdgeDegree ( ) const

inline

Return the degree of edge polynomials.

ElementQuad()

ElementQuad::ElementQuad	(	const HybridCore &	hho,
		const size_t	iT,
		const size_t	doeT,
		const size_t	doeF
	)

Class constructor: loads the quadrature rules and values of basis functions/gradients at these points.

Parameters

hho	A reference to the hybridcore instance
iT	Global index of cell
doeT	The degree of exactness for cell quadratures
doeF	The degree of exactness of edge quadratures

evaluate_in_cell()

double HybridCore::evaluate_in_cell	(	const UVector	Xh,
		size_t	iT,
		VectorRd	x
	)		const

Evaluates a discrete function in the cell iT at point x.

evaluate_in_edge()

double HybridCore::evaluate_in_edge	(	const UVector	Xh,
		size_t	iE,
		VectorRd	x
	)		const

Evaluates a discrete function on the edge iE at point x.

get_cell_deg()

const int HArDCore2D::UVector::get_cell_deg ( ) const

inline

Return the cell degree.

get_dphiT_quadF()

const boost::multi_array<VectorRd, 2>& HArDCore2D::ElementQuad::get_dphiT_quadF ( size_t  ilF ) const

inline

Returns values of gradients of cell basis functions at edge quadrature nodes, for edge with local number ilF.

get_dphiT_quadT()

const boost::multi_array<VectorRd, 2>& HArDCore2D::ElementQuad::get_dphiT_quadT ( ) const

inline

Returns values of gradients of cell basis functions at cell quadrature nodes.

get_edge_deg()

const size_t HArDCore2D::UVector::get_edge_deg ( ) const

inline

Return the edge degree.

get_mesh()

const Mesh* HArDCore2D::HybridCore::get_mesh ( ) const

inline

Returns a pointer to the mesh.

get_phiF_quadF()

const boost::multi_array<double, 2> HArDCore2D::ElementQuad::get_phiF_quadF ( size_t  ilF ) const

inline

Returns values of edge basis functions at edge quadrature nodes, for edge with local number ilF.

get_phiT_quadF()

const boost::multi_array<double, 2>& HArDCore2D::ElementQuad::get_phiT_quadF ( size_t  ilF ) const

inline

Returns values of cell basis functions at edge quadrature nodes, for edge with local number ilF.

get_phiT_quadT()

const boost::multi_array<double, 2>& HArDCore2D::ElementQuad::get_phiT_quadT ( ) const

inline

Returns values of cell basis functions at cell quadrature nodes.

get_quadF()

const QuadratureRule& HArDCore2D::ElementQuad::get_quadF ( size_t  ilF ) const

inline

Returns quadrature rules on edge with local number ilF.

get_quadT()

const QuadratureRule& HArDCore2D::ElementQuad::get_quadT ( ) const

inline

Returns quadrature rules in cell.

get_vec_phiF_quadF()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiF_quadF	(	size_t	ilF,
		size_t	degree
	)		const

Builds on the fly the values of vector edge basis functions at edge quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

local number of edge required degree of basis function

get_vec_phiT_quadF()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiT_quadF	(	size_t	ilF,
		size_t	degree
	)		const

Builds on the fly the values of vector cell basis functions at edge quadrature nodes. The vector basis is obtained by tensorization of the scalar one as in get_vec_phiT_quadT.

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

local number of edge maximum degree of basis functions required

get_vec_phiT_quadT()

boost::multi_array< VectorRd, 2 > ElementQuad::get_vec_phiT_quadT

(

size_t

degree

)

const

Builds on the fly the values of vector cell basis functions at cell quadrature nodes. The vector basis is obtained by tensorization of the scalar one: \((\phi_1,0), (\phi_2,0), ..., (\phi_N,0), (0,\phi_1), (0,\phi_2) ... (0,\phi_N)\).

Returns

vec_phiT_quadT such that, for r=0,..,dim-1 and i=0,..,DimPoly<Cell>(degree)-1, vec_phiT_quadT[r*DimPoly<Cell>(degree)-1 + i] has, on its row r, the values of the i-th scalar basis function at the quadrature nodes, and 0 on its other rows.

Parameters

degree

maximal degree of basis functions that is required

H1norm()

double HybridCore::H1norm

(

const UVector &

Xh

)

const

Compute discrete H1 norm of a discrete function.

Parameters

Xh	Vector of unknowns

HybridCore()

HybridCore::HybridCore	(	const Mesh *	mesh_ptr,
		const int	cell_deg,
		const size_t	edge_deg,
		const bool	use_threads = true,
		std::ostream &	output = std::cout,
		const bool	ortho = true
	)

Class constructor: initialises the data structure with the given mesh, and desired polynomial degrees of the basis functions.

The orthonormalisation comes at a cost in terms of manipulation of the basis functions. This should only be used when the polynomial degree is large and/or the cell is distorted. However, in these cases, it can make a huge difference on the observed convergence rate.

Parameters

mesh_ptr	A pointer to the loaded mesh
cell_deg	The degree of the cell polynomials
edge_deg	The degree of the edge polynomials
use_threads	Optional argument to indicate if threads should be used
output	Optional argument for specifying outputs of messages
ortho	Optional argument for choosing to orthonormalise or not the basis functions

interpolate()

template<typename ContinuousFunction >

UVector HArDCore2D::HybridCore::interpolate	(	const ContinuousFunction &	f,
		const int	deg_cell,
		const size_t	deg_edge,
		size_t	doe
	)		const

Compute the interpolant in the discrete space of a continuous function.

Returns

vector Xh of coefficients on the basis functions, as described in class UVector.

Parameters

f	function to interpolate
deg_cell	degree of cell polynomials for interpolation
deg_edge	degree of edge polynomials for interpolation
doe	degree of exactness of the quadrature rules to compute the interpolate

L2norm()

double HybridCore::L2norm

(

const UVector &

Xh

)

const

Compute L2 norm of a discrete function (using cell values)

Parameters

Xh	Vector of unknowns

n_cell_dofs()

const size_t HArDCore2D::UVector::n_cell_dofs ( ) const

inline

Number of dofs in each cell.

n_edge_dofs()

const size_t HArDCore2D::UVector::n_edge_dofs ( ) const

inline

Number of dofs on each edge.

n_total_cell_dofs()

const size_t HArDCore2D::UVector::n_total_cell_dofs ( ) const

inline

Total number of cell dofs (in the vector, this is where the edge dofs start)

operator()()

double HArDCore2D::UVector::operator() ( size_t  index ) const

inline

Overloads the (): returns the corresponding coefficient.

operator+()

UVector HArDCore2D::UVector::operator+ ( const UVector &  b )

inline

Overloads the addition: adds the coefficients.

operator+=()

void HArDCore2D::UVector::operator+= ( const UVector &  b )

inline

Overloads the increment: adds the coefficients.

operator-()

UVector HArDCore2D::UVector::operator- ( const UVector &  b )

inline

Overloads the subtraction: subtracts the coefficients.

operator/=()

void HArDCore2D::UVector::operator/= ( const double &  r )

inline

Overloads the increment: adds the coefficients.

restr()

Eigen::VectorXd UVector::restr

(

size_t

iT

)

const

Extract the restriction of the unknowns corresponding to cell iT and its edges.

UVector()

UVector::UVector	(	const Eigen::VectorXd &	values,
		const Mesh &	mesh,
		const int	cell_deg,
		const size_t	edge_deg
	)

Parameters

values	values of the vector
mesh	reference to the mesh
cell_deg	polynomial degrees in cell
edge_deg	polynomial degrees on edge

VertexValues()

Eigen::VectorXd HybridCore::VertexValues	(	const UVector	Xh,
		const std::string	from_dofs
	)

From a hybrid function, computes a vector of values at the vertices of the mesh.

Parameters

Xh	vector of discrete unknowns on cell and edge polynomials
from_dofs	Type of unknowns to use: "cell" or "edge"