Reference documentation for deal.II version GIT 9fcc068bd8 20230606 14:45:02+00:00

#include <deal.II/base/polynomials_bernardi_raugel.h>
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 
This class implements the BernardiRaugel polynomials similarly to the description in the Mathematics of Computation paper from 1985 by Christine Bernardi and GeneviÃ¨ve Raugel.
The BernardiRaugel 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 lockingfree 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.
@f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1y)@f$ @f$y=0@f$ edge: @f$\mathbf{p}_3 = \mathbf{n}_3 (x)(1x)(1y)@f$ @f$y=1@f$ edge: @f$\mathbf{p}_4 = \mathbf{n}_4 (x)(1x)(y)@f$
@f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1y)(z)(1z)@f$ @f$y=0@f$ edge: @f$\mathbf{p}_3 = \mathbf{n}_3 (x)(1x)(1y)(z)(1z)@f$ @f$y=1@f$ edge: @f$\mathbf{p}_4 = \mathbf{n}_4 (x)(1x)(y)(z)(1z)@f$ @f$z=0@f$ edge: @f$\mathbf{p}_5 = \mathbf{n}_5 (x)(1x)(y)(1y)(1z)@f$ @f$z=1@f$ edge: @f$\mathbf{p}_6 = \mathbf{n}_6 (x)(1x)(y)(1y)(z)@f$
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< dim >::PolynomialsBernardiRaugel  (  const unsigned int  k  ) 
Constructor. Creates all basis functions for BernardiRaugel polynomials of given degree.
k=1
. Definition at line 25 of file polynomials_bernardi_raugel.cc.

inlineoverridevirtual 
Return the name of the space, which is BernardiRaugel
.
Implements TensorPolynomialsBase< dim >.
Definition at line 171 of file polynomials_bernardi_raugel.h.

overridevirtual 
Compute the value and derivatives of each BernardiRaugel 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.

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.

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.

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.

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.

inlineinherited 
Return the number of polynomials.
Definition at line 157 of file tensor_polynomials_base.h.

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 166 of file tensor_polynomials_base.h.

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.

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.

privateinherited 
The highest polynomial degree of this functions represented by this object.
Definition at line 145 of file tensor_polynomials_base.h.

privateinherited 
The number of polynomials represented by this object.
Definition at line 150 of file tensor_polynomials_base.h.