Implementation of the locally H(grad)-compatible FEM basis of variable order on the [-1,1] reference line cell, using Jacobi polynomials. More...

#include <Intrepid_HGRAD_LINE_Cn_FEM_JACOBI.hpp>

Inheritance diagram for Intrepid::Basis_HGRAD_LINE_Cn_FEM_JACOBI< Scalar, ArrayScalar >:

Public Member Functions
	Basis_HGRAD_LINE_Cn_FEM_JACOBI (int order, Scalar alpha=0, Scalar beta=0)
	Constructor.

void	getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
	Evaluation of a FEM basis on a reference Line cell. More...

void	getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const ArrayScalar &cellVertices, const EOperator operatorType=OPERATOR_VALUE) const
	FVD basis evaluation: invocation of this method throws an exception.

void	setBasisParameters (int n, Scalar alpha=0, Scalar beta=0)
	Sets private data basisDegree_, basisCardinality_, jacobiAlpha_, and jacobiBeta_, to n, n+1, alpha, and beta, respectively.

Public Member Functions inherited from Intrepid::Basis< Scalar, ArrayScalar >
virtual	~Basis ()
	Destructor.

virtual int	getCardinality () const
	Returns cardinality of the basis. More...

virtual int	getDegree () const
	Returns the degree of the basis. More...

virtual const shards::CellTopology	getBaseCellTopology () const
	Returns the base cell topology for which the basis is defined. See Shards documentation http://trilinos.sandia.gov/packages/shards for definition of base cell topology. More...

virtual EBasis	getBasisType () const
	Returns the basis type. More...

virtual ECoordinates	getCoordinateSystem () const
	Returns the type of coordinate system for which the basis is defined. More...

virtual int	getDofOrdinal (const int subcDim, const int subcOrd, const int subcDofOrd)
	DoF tag to ordinal lookup. More...

virtual const std::vector < std::vector< std::vector < int > > > &	getDofOrdinalData ()
	DoF tag to ordinal data structure.

virtual const std::vector< int > &	getDofTag (const int dofOrd)
	DoF ordinal to DoF tag lookup. More...

virtual const std::vector < std::vector< int > > &	getAllDofTags ()
	Retrieves all DoF tags. More...

Private Member Functions
void	initializeTags ()
	Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.

Private Attributes
Scalar	jacobiAlpha_

Scalar	jacobiBeta_

Additional Inherited Members
Protected Attributes inherited from Intrepid::Basis< Scalar, ArrayScalar >
int	basisCardinality_
	Cardinality of the basis, i.e., the number of basis functions/degrees-of-freedom.

int	basisDegree_
	Degree of the largest complete polynomial space that can be represented by the basis.

shards::CellTopology	basisCellTopology_
	Base topology of the cells for which the basis is defined. See the Shards package http://trilinos.sandia.gov/packages/shards for definition of base cell topology.

EBasis	basisType_
	Type of the basis.

ECoordinates	basisCoordinates_
	The coordinate system for which the basis is defined.

bool	basisTagsAreSet_
	"true" if tagToOrdinal_ and ordinalToTag_ have been initialized

std::vector< std::vector< int > >	ordinalToTag_
	DoF ordinal to tag lookup table. More...

std::vector< std::vector < std::vector< int > > >	tagToOrdinal_
	DoF tag to ordinal lookup table. More...

Detailed Description

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HGRAD_LINE_Cn_FEM_JACOBI< Scalar, ArrayScalar >

Implementation of the locally H(grad)-compatible FEM basis of variable order on the [-1,1] reference line cell, using Jacobi polynomials.

      Implements Jacobi basis of variable order \form#162 on
      the reference [-1,1] line cell. Jacobi polynomials depend on three parameters

$\alpha$ , $\beta$ , and $ n $ and are defined via the so-called Gamma function by

$P_n^{(\alpha,\beta)} (z) = \frac{\Gamma (\alpha+n+1)}{n!\Gamma (\alpha+\beta+n+1)} \sum_{m=0}^n {n\choose m} \frac{\Gamma (\alpha + \beta + n + m + 1)}{\Gamma (\alpha + m + 1)} \left(\frac{z-1}{2}\right)^m$

The basis has cardinality $n+1$ and spans a COMPLETE linear polynomial space. Basis functions are dual to a unisolvent set of degrees of freedom (DoF) enumerated as follows:

Basis order	DoF tag table				DoF definition
Basis order	subc dim	subc ordinal	subc DoF tag	subc num DoFs	DoF definition
0	1	0	0	1	$P_0^{(\alpha,\beta)}$
1	1	0	0-1	2	$P_0^{(\alpha,\beta)}, P_1^{(\alpha,\beta)}$
2	1	0	0-2	3	$P_0^{(\alpha,\beta)}, P_1^{(\alpha,\beta)}, P_2^{(\alpha,\beta)}$
3	1	0	0-3	4	$P_0^{(\alpha,\beta)}, P_1^{(\alpha,\beta)}, ..., P_3^{(\alpha,\beta)}$
...	1	0	...	...	...
n	1	0	0-n	n+1	$P_0^{(\alpha,\beta)}, P_1^{(\alpha,\beta)}, ..., P_n^{(\alpha,\beta)}$

      For example, for Legendre polynomials ( \form#173), the first 11 bases are given by

Basis order	DoF tag table				DoF definition
Basis order	subc dim	subc ordinal	subc DoF tag	subc num DoFs	DoF definition
0	1	0	0	1
1	1	0	0-1	2	and:
2	1	0	0-2	3	and: $\frac{1}{2} (3x^2-1)$
3	1	0	0-3	4	and: $\frac{1}{2} (5x^3-3x)$
4	1	0	0-4	5	and: $\frac{1}{8} (35x^4-30x^2+3)$
5	1	0	0-5	6	and: $\frac{1}{8} (63x^5-70x^3+15x)$
6	1	0	0-6	7	and: $\frac{1}{16} (231x^6-315x^4+105x^2-5)$
7	1	0	0-7	8	and: $\frac{1}{16} (429x^7-693x^5+315x^3-35x)$
8	1	0	0-8	9	and: $\frac{1}{128} (6435x^8-12012x^6+6930x^4-1260x^2+35)$
9	1	0	0-9	10	and: $\frac{1}{128} (12155x^9-25740x^7+18018x^5-4620x^3+315x)$
10	1	0	0-10	11	and: $\frac{1}{128} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)$

Definition at line 131 of file Intrepid_HGRAD_LINE_Cn_FEM_JACOBI.hpp.

Member Function Documentation

template<class Scalar , class ArrayScalar>

void Intrepid::Basis_HGRAD_LINE_Cn_FEM_JACOBI< Scalar, ArrayScalar >::getValues	(	ArrayScalar &	outputValues,
		const ArrayScalar &	inputPoints,
		const EOperator	operatorType
	)		const

virtual

Evaluation of a FEM basis on a reference Line cell.

    Returns values of <var>operatorType</var> acting on FEM basis functions for a set of
    points in the <strong>reference Line</strong> cell. For rank and dimensions of
    I/O array arguments see Section \ref basis_md_array_sec .

Parameters

outputValues	[out] - variable rank array with the basis values
inputPoints	[in] - rank-2 array (P,D) with the evaluation points
operatorType	[in] - the operator acting on the basis functions

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 69 of file Intrepid_HGRAD_LINE_Cn_FEM_JACOBIDef.hpp.

References Intrepid::IntrepidPolylib::jacobd(), and Intrepid::IntrepidPolylib::jacobfd().

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

http://docs.trilinos.org/r12.18/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Cn_FEM_JACOBI.hpp
http://docs.trilinos.org/r12.18/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Cn_FEM_JACOBIDef.hpp

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<class Scalar, class ArrayScalar> class Intrepid::Basis_HGRAD_LINE_Cn_FEM_JACOBI< Scalar, ArrayScalar >

Member Function Documentation

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HGRAD_LINE_Cn_FEM_JACOBI< Scalar, ArrayScalar >