Reference documentation for deal.II version 9.4.1
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PolynomialsRT_Bubbles< dim > Class Template Reference

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

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

PolynomialsRT_Bubbles (const unsigned int k)

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

std::string name () 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 degree)

## Private Attributes

const PolynomialsRaviartThomas< dim > raviart_thomas_space

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

const unsigned int polynomial_degree

const 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: 45 46 47 48 49

\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: 53 54 55 56 57 58 59 60 61 62 63 64

\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 - $$3D,\,k=2$$.

Definition at line 86 of file polynomials_rt_bubbles.h.

## ◆ PolynomialsRT_Bubbles()

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 30 of file polynomials_rt_bubbles.cc.

## ◆ evaluate()

template<int dim>
 void PolynomialsRT_Bubbles< 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

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.

Implements TensorPolynomialsBase< dim >.

Definition at line 45 of file polynomials_rt_bubbles.cc.

## ◆ name()

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

Return the name of the space, which is RT_Bubbles.

Implements TensorPolynomialsBase< dim >.

Definition at line 151 of file polynomials_rt_bubbles.h.

## ◆ n_polynomials()

template<int dim>
 unsigned int PolynomialsRT_Bubbles< dim >::n_polynomials ( 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 838 of file polynomials_rt_bubbles.cc.

## ◆ clone()

template<int dim>
 std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsRT_Bubbles< 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 850 of file polynomials_rt_bubbles.cc.

## ◆ n()

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

Return the number of polynomials.

Definition at line 157 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 166 of file tensor_polynomials_base.h.

## ◆ raviart_thomas_space

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 139 of file polynomials_rt_bubbles.h.

## ◆ monomials

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 145 of file polynomials_rt_bubbles.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 145 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 150 of file tensor_polynomials_base.h.

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