Reference documentation for deal.II version 9.1.1
PolynomialsBernardiRaugel< dim > Class Template Reference

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

## Public Member Functions

PolynomialsBernardiRaugel (const unsigned int k)

unsigned int n () const

unsigned int degree () const

std::string name () const

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

## Static Public Member Functions

static unsigned int compute_n_pols (const unsigned int k)

## Static Private Member Functions

static std::vector< std::vector< Polynomials::Polynomial< double > > > create_polynomials_Q ()

static std::vector< std::vector< Polynomials::Polynomial< double > > > create_polynomials_bubble ()

## Private Attributes

const unsigned int my_degree

const unsigned int n_pols

const AnisotropicPolynomials< dim > polynomial_space_Q

const AnisotropicPolynomials< dim > polynomial_space_bubble

## Detailed Description

### template<int dim> class PolynomialsBernardiRaugel< dim >

This class implements the Bernardi-Raugel polynomials similarly to the description in the Mathematics of Computation paper from 1985 by Christine Bernardi and Geneviève Raugel.

The Bernardi-Raugel polynomials are originally defined as an enrichment of the $$(P_1)^d$$ elements on simplicial meshes for Stokes problems by the addition of bubble functions, yielding a locking-free finite element which is a subset of $$(P_2)^d$$ elements. This implementation is an enrichment of $$(Q_1)^d$$ elements which is a subset of $$(Q_2)^d$$ elements for quadrilateral and hexahedral meshes.

The $$BR_1$$ bubble functions are defined to have magnitude 1 at the center of face $$e_i$$ and direction $$\mathbf{n}_i$$ normal to face $$e_i$$, and magnitude 0 on all other vertices and faces. Ordering is consistent with the face numbering in GeometryInfo. The vector $$\mathbf{n}_i$$ points in the positive axis direction and not necessarily normal to the element for consistent orientation across edges.

2D bubble functions (in order)

$$x=0$$ edge: $$\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)$$

$$x=1$$ edge: $$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)$$

$$y=0$$ edge: $$\mathbf{p}_3 = \mathbf{n}_3 (x)(1-x)(1-y)$$

$$y=1$$ edge: $$\mathbf{p}_4 = \mathbf{n}_4 (x)(1-x)(y)$$

3D bubble functions (in order)

$$x=0$$ edge: $$\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)(z)(1-z)$$

$$x=1$$ edge: $$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)(z)(1-z)$$

$$y=0$$ edge: $$\mathbf{p}_3 = \mathbf{n}_3 (x)(1-x)(1-y)(z)(1-z)$$

$$y=1$$ edge: $$\mathbf{p}_4 = \mathbf{n}_4 (x)(1-x)(y)(z)(1-z)$$

$$z=0$$ edge: $$\mathbf{p}_5 = \mathbf{n}_5 (x)(1-x)(y)(1-y)(1-z)$$

$$z=1$$ edge: $$\mathbf{p}_6 = \mathbf{n}_6 (x)(1-x)(y)(1-y)(z)$$

Then the $$BR_1(E)$$ polynomials are defined on quadrilaterals and hexahedra by $$BR_1(E) = Q_1(E) \oplus \mbox{span}\{\mathbf{p}_i, i=1,...,2d\}$$.

Date
2018

Definition at line 84 of file polynomials_bernardi_raugel.h.

## ◆ PolynomialsBernardiRaugel()

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

Constructor. Creates all basis functions for Bernardi-Raugel polynomials of given degree.

• k: the degree of the Bernardi-Raugel-space, which is currently limited to the case k=1

Definition at line 23 of file polynomials_bernardi_raugel.cc.

## ◆ n()

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

Return the number of Bernardi-Raugel polynomials.

Definition at line 189 of file polynomials_bernardi_raugel.h.

## ◆ degree()

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

Return the degree of Bernardi-Raugel polynomials. Since the bubble functions are quadratic in at least one variable, the degree of the Bernardi-Raugel polynomials is two.

Definition at line 198 of file polynomials_bernardi_raugel.h.

## ◆ name()

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

Return the name of the space, which is BernardiRaugel.

Definition at line 207 of file polynomials_bernardi_raugel.h.

## ◆ compute()

template<int dim>
 void PolynomialsBernardiRaugel< 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 derivatives of each Bernardi- Raugel 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 70 of file polynomials_bernardi_raugel.cc.

## ◆ compute_n_pols()

template<int dim>
 unsigned int PolynomialsBernardiRaugel< dim >::compute_n_pols ( const unsigned int k )
static

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

Definition at line 238 of file polynomials_bernardi_raugel.cc.

## ◆ create_polynomials_Q()

template<int dim>
 std::vector< std::vector< Polynomials::Polynomial< double > > > PolynomialsBernardiRaugel< dim >::create_polynomials_Q ( )
staticprivate

A static member function that creates the polynomial space we use to initialize the polynomial_space_Q member variable.

Definition at line 55 of file polynomials_bernardi_raugel.cc.

## ◆ create_polynomials_bubble()

template<int dim>
 std::vector< std::vector< Polynomials::Polynomial< double > > > PolynomialsBernardiRaugel< dim >::create_polynomials_bubble ( )
staticprivate

A static member function that creates the polynomial space we use to initialize the polynomial_space_bubble member variable.

Definition at line 33 of file polynomials_bernardi_raugel.cc.

## ◆ my_degree

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

The degree of this object given to the constructor (must be 1).

Definition at line 149 of file polynomials_bernardi_raugel.h.

## ◆ n_pols

template<int dim>
 const unsigned int PolynomialsBernardiRaugel< dim >::n_pols
private

The number of Bernardi-Raugel polynomials.

Definition at line 154 of file polynomials_bernardi_raugel.h.

## ◆ polynomial_space_Q

template<int dim>
 const AnisotropicPolynomials PolynomialsBernardiRaugel< dim >::polynomial_space_Q
private

An object representing the polynomial space of Q functions which forms the BR polynomials through outer products of these with the corresponding unit ijk vectors.

Definition at line 162 of file polynomials_bernardi_raugel.h.

## ◆ polynomial_space_bubble

template<int dim>
 const AnisotropicPolynomials PolynomialsBernardiRaugel< dim >::polynomial_space_bubble
private

An object representing the polynomial space of bubble functions which forms the BR polynomials through outer products of these with the corresponding normals.

Definition at line 169 of file polynomials_bernardi_raugel.h.

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