Reference documentation for deal.II version 8.5.0
PolynomialsABF< dim > Class Template Reference

#include <deal.II/base/polynomials_abf.h>

## Public Member Functions

PolynomialsABF (const unsigned int k)

void compute (const Point< dim > &unit_point, std::vector< Tensor< 1, dim > > &values, std::vector< Tensor< 2, dim > > &grads, std::vector< Tensor< 3, dim > > &grad_grads, std::vector< Tensor< 4, dim > > &third_derivatives, std::vector< Tensor< 5, dim > > &fourth_derivatives) const

unsigned int n () const

unsigned int degree () const

std::string name () const

## Static Public Member Functions

static unsigned int compute_n_pols (unsigned int degree)

## Private Attributes

const unsigned int my_degree

const AnisotropicPolynomials< dim > polynomial_space

unsigned int n_pols

std::vector< double > p_values

std::vector< Tensor< 1, dim > > p_grads

std::vector< Tensor< 3, dim > > p_third_derivatives

std::vector< Tensor< 4, dim > > p_fourth_derivatives

## Detailed Description

### template<int dim> class PolynomialsABF< dim >

This class implements the Hdiv-conforming, vector-valued Arnold-Boffi-Falk polynomials as described in the article by Arnold-Boffi- Falk: Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. Vol.42, No.6, pp.2429-2451

The ABF polynomials are constructed such that the divergence is in the tensor product polynomial space Qk. Therefore, the polynomial order of each component must be two orders higher in the corresponding direction, yielding the polynomial spaces (Qk+2,k, Qk,k+2) and (Qk+2,k,k, Qk,k+2,k, Qk,k,k+2) in 2D and 3D, resp.

Date
2006

Definition at line 54 of file polynomials_abf.h.

## Constructor & Destructor Documentation

template<int dim>
 PolynomialsABF< dim >::PolynomialsABF ( const unsigned int k )

Constructor. Creates all basis functions for Raviart-Thomas polynomials of given degree.

• k: the degree of the Raviart-Thomas-space, which is the degree of the largest tensor product polynomial space Qk contained.

Definition at line 48 of file polynomials_abf.cc.

## Member Function Documentation

template<int dim>
 void PolynomialsABF< dim >::compute ( const Point< dim > & unit_point, std::vector< Tensor< 1, dim > > & values, std::vector< Tensor< 2, dim > > & grads, std::vector< Tensor< 3, dim > > & grad_grads, std::vector< Tensor< 4, dim > > & third_derivatives, std::vector< Tensor< 5, dim > > & fourth_derivatives ) const

Compute the value and the first and second derivatives of each Raviart- Thomas polynomial at unit_point.

The size of the vectors must either be zero or equal n(). In the first case, the function will not compute these values.

If you need values or derivatives of all tensor product polynomials then use this function, rather than using any of the compute_value, compute_grad or compute_grad_grad functions, see below, in a loop over all tensor product polynomials.

Definition at line 65 of file polynomials_abf.cc.

template<int dim>
 unsigned int PolynomialsABF< dim >::n ( ) const
inline

Return the number of ABF polynomials.

Definition at line 161 of file polynomials_abf.h.

template<int dim>
 unsigned int PolynomialsABF< dim >::degree ( ) const
inline

Return the degree of the ABF space, which is two less than the highest polynomial degree.

Definition at line 169 of file polynomials_abf.h.

template<int dim>
 std::string PolynomialsABF< dim >::name ( ) const
inline

Return the name of the space, which is ABF.

Definition at line 177 of file polynomials_abf.h.

template<int dim>
 unsigned int PolynomialsABF< dim >::compute_n_pols ( unsigned int degree )
static

Return the number of polynomials in the space RT(degree) without requiring to build an object of PolynomialsABF. This is required by the FiniteElement classes.

Definition at line 150 of file polynomials_abf.cc.

## Member Data Documentation

template<int dim>
 const unsigned int PolynomialsABF< dim >::my_degree
private

The degree of this object as given to the constructor.

Definition at line 113 of file polynomials_abf.h.

template<int dim>
 const AnisotropicPolynomials PolynomialsABF< dim >::polynomial_space
private

An object representing the polynomial space for a single component. We can re-use it for the other vector components by rotating the coordinates of the evaluation point.

Definition at line 120 of file polynomials_abf.h.

template<int dim>
 unsigned int PolynomialsABF< dim >::n_pols
private

Number of Raviart-Thomas polynomials.

Definition at line 125 of file polynomials_abf.h.

template<int dim>
mutableprivate

A mutex that guards the following scratch arrays.

Definition at line 130 of file polynomials_abf.h.

template<int dim>
 std::vector PolynomialsABF< dim >::p_values
mutableprivate

Auxiliary memory.

Definition at line 135 of file polynomials_abf.h.

template<int dim>
mutableprivate

Auxiliary memory.

Definition at line 140 of file polynomials_abf.h.

template<int dim>
mutableprivate

Auxiliary memory.

Definition at line 145 of file polynomials_abf.h.

template<int dim>
 std::vector > PolynomialsABF< dim >::p_third_derivatives
mutableprivate

Auxiliary memory.

Definition at line 150 of file polynomials_abf.h.

template<int dim>
 std::vector > PolynomialsABF< dim >::p_fourth_derivatives
mutableprivate

Auxiliary memory.

Definition at line 155 of file polynomials_abf.h.

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