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 >:

List of all members.

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.
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.
Private Member Functions
void	initializeTags ()
	Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.
Private Attributes
Scalar	jacobiAlpha_
Scalar	jacobiBeta_

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 $n$ 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 ( $\alpha=\beta=0$ ), 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 operatorType acting on FEM basis functions for a set of points in the reference Line cell. For rank and dimensions of I/O array arguments see Section MD array template arguments for basis methods .

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().

Referenced by Intrepid::Basis_HGRAD_LINE_Cn_FEM< Scalar, ArrayScalar >::Basis_HGRAD_LINE_Cn_FEM().

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

/usr/src/RPM/BUILD/trilinos-11.12.1/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Cn_FEM_JACOBI.hpp
/usr/src/RPM/BUILD/trilinos-11.12.1/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Cn_FEM_JACOBIDef.hpp

Public Member Functions

Private Member Functions

Private Attributes

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 >