Reference documentation for deal.II version 9.3.3
PolynomialsBernardiRaugel< dim > Class Template Reference

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

Inheritance diagram for PolynomialsBernardiRaugel< dim >:
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## Public Member Functions

PolynomialsBernardiRaugel (const unsigned int k)

std::string name () const override

void evaluate (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 override

virtual std::unique_ptr< TensorPolynomialsBase< dim > > clone () const override

unsigned int n () const

unsigned int degree () const

## Static Public Member Functions

static unsigned int n_polynomials (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 AnisotropicPolynomials< dim > polynomial_space_Q

const AnisotropicPolynomials< dim > polynomial_space_bubble

const unsigned int polynomial_degree

const unsigned int n_pols

## 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: $$\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)$$
 @f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)@f$

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

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

3D bubble functions (in order)
$$x=0$$ edge: $$\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)(z)(1-z)$$
 @f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)(z)(1-z)@f$

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

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

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

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


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

Definition at line 85 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 25 of file polynomials_bernardi_raugel.cc.

## ◆ name()

template<int dim>
 std::string PolynomialsBernardiRaugel< dim >::name
inlineoverridevirtual

Return the name of the space, which is BernardiRaugel.

Implements TensorPolynomialsBase< dim >.

Definition at line 171 of file polynomials_bernardi_raugel.h.

## ◆ evaluate()

template<int dim>
 void PolynomialsBernardiRaugel< dim >::evaluate ( 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
overridevirtual

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.

Implements TensorPolynomialsBase< dim >.

Definition at line 71 of file polynomials_bernardi_raugel.cc.

## ◆ n_polynomials()

template<int dim>
 unsigned int PolynomialsBernardiRaugel< dim >::n_polynomials ( 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 240 of file polynomials_bernardi_raugel.cc.

## ◆ clone()

template<int dim>
 std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsBernardiRaugel< dim >::clone
overridevirtual

A sort of virtual copy constructor, this function returns a copy of the polynomial space object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.

Some places in the library, for example the constructors of FE_PolyTensor, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.

Implements TensorPolynomialsBase< dim >.

Definition at line 256 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 56 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 34 of file polynomials_bernardi_raugel.cc.

## ◆ n()

template<int dim>
 unsigned int TensorPolynomialsBase< dim >::n
inlineinherited

Return the number of polynomials.

Definition at line 155 of file tensor_polynomials_base.h.

## ◆ degree()

template<int dim>
 unsigned int TensorPolynomialsBase< dim >::degree
inlineinherited

Return the highest polynomial degree of polynomials represented by this class. A derived class may override this if its value is different from my_degree.

Definition at line 164 of file tensor_polynomials_base.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 144 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 151 of file polynomials_bernardi_raugel.h.

## ◆ polynomial_degree

template<int dim>
 const unsigned int TensorPolynomialsBase< dim >::polynomial_degree
privateinherited

The highest polynomial degree of this functions represented by this object.

Definition at line 143 of file tensor_polynomials_base.h.

## ◆ n_pols

template<int dim>
 const unsigned int TensorPolynomialsBase< dim >::n_pols
privateinherited

The number of polynomials represented by this object.

Definition at line 148 of file tensor_polynomials_base.h.

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