Reference documentation for deal.II version 9.0.0
PolynomialsRT_Bubbles< dim > Class Template Reference

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

## Public Member Functions

PolynomialsRT_Bubbles (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 (const unsigned int degree)

## Private Attributes

const unsigned int my_degree

const PolynomialsRaviartThomas< dim > raviart_thomas_space

std::vector< Polynomials::Polynomial< double > > monomials

unsigned int n_pols

## Detailed Description

### template<int dim> class PolynomialsRT_Bubbles< dim >

This class implements the Hdiv-conforming, vector-valued enhanced Raviart-Thomas polynomials.

Similarly to the classical Raviart-Thomas space, the enhanced Raviart-Thomas polynomials are constructed such that the divergence is in the tensor product polynomial space Qk-1.

This space is of the form Vk = RTk-1 + Bk, where Bk is defined as follows:

In 2D:

\begin{align*} B_k^1(E) = \text{span}\left\{x^{a_1-1} y^{a_2}\begin{pmatrix} (a_2+1) x \\ -a_1 y \end{pmatrix}\text{ : } a_2=k \right\} \\ B_k^2(E) = \text{span}\left\{x^{b_1} y^{b_2-1}\begin{pmatrix} -b_2 x \\ (b_1+1) y \end{pmatrix}\text{ : } b_1=k \right\} \end{align*}

In 3D:

\begin{align*} B_k^1(E) = \text{span}\left\{x^{a_1-1} y^{a_2} z^{a_3}\begin{pmatrix} (a_2+a_3+2) x \\ -a_1 y \\ -a_1 z \end{pmatrix}\text{ : } a_2=k \text{ or } a_3=k \right\},\\ B_k^2(E) = \text{span}\left\{x^{b_1} y^{b_2-1} z^{b_3}\begin{pmatrix} -b_2 x \\ (b_1+b_3+2) y \\ -b_2 z \end{pmatrix}\text{ : } b_1=k \text{ or } b_3=k \right\},\\ B_k^3(E) = \text{span}\left\{x^{c_1}y^{c_2}z^{c_3-1}\begin{pmatrix} -c_3 x \\ -c_3y \\ (c_1+c_2+2)z \end{pmatrix}\text{ : } c_1=k \text{ or } c_2=k \right\}, \end{align*}

where $$0 \le a_1, a_2, a_3 \le k$$.

Note
Unlike the classical Raviart-Thomas space, the lowest order for the enhanced space is 1, similarly to the Brezzi-Douglas-Marini (BDM) polynomial space.

The total dimension of the space dim(Vk) = d*(k+1)^d, where d is the space dimension. This allows to associate shape functions with the Gauss-Lobatto quadrature points as shown in the figures below.

 Left - $$2D,\,k=3$$, right - $$3D,\,k=2$$.
Date
2018

Definition at line 83 of file polynomials_rt_bubbles.h.

## Constructor & Destructor Documentation

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

Constructor. Creates all basis functions for RT_bubbles polynomials of given degree.

Definition at line 27 of file polynomials_rt_bubbles.cc.

## Member Function Documentation

template<int dim>
 void PolynomialsRT_Bubbles< 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

Computes the value and the first and second derivatives of each RT_bubbles 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 44 of file polynomials_rt_bubbles.cc.

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

Returns the number of enhanced Raviart-Thomas polynomials.

Definition at line 161 of file polynomials_rt_bubbles.h.

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

Returns the degree of the RT_bubble space, which is one less than the highest polynomial degree.

Definition at line 170 of file polynomials_rt_bubbles.h.

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

Return the name of the space, which is RT_Bubbles.

Definition at line 179 of file polynomials_rt_bubbles.h.

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

Return the number of polynomials in the space RT_Bubbles(degree) without requiring to build an object of PolynomialsRT-Bubbles. This is required by the FiniteElement classes.

Definition at line 493 of file polynomials_rt_bubbles.cc.

## Member Data Documentation

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

The degree of this object as given to the constructor.

Definition at line 138 of file polynomials_rt_bubbles.h.

template<int dim>
 const PolynomialsRaviartThomas PolynomialsRT_Bubbles< dim >::raviart_thomas_space
private

An object representing the Raviart-Thomas part of the space

Definition at line 143 of file polynomials_rt_bubbles.h.

template<int dim>
 std::vector > PolynomialsRT_Bubbles< dim >::monomials
private

Storage for monomials, we need all polynomials from degree zero to k+1.

Definition at line 149 of file polynomials_rt_bubbles.h.

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

Number of RT_Bubbles polynomials.

Definition at line 154 of file polynomials_rt_bubbles.h.

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