Reference documentation for deal.II version Git f15f581df6 2020-07-10 15:30:09 -0400
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Public Member Functions | Static Public Member Functions | Private Attributes | List of all members
PolynomialsRT_Bubbles< dim > Class Template Reference

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

Inheritance diagram for PolynomialsRT_Bubbles< dim >:

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
- Public Member Functions inherited from TensorPolynomialsBase< dim >
 TensorPolynomialsBase (const unsigned int deg, const unsigned int n_polynomials)
 TensorPolynomialsBase (TensorPolynomialsBase< dim > &&)=default
 TensorPolynomialsBase (const TensorPolynomialsBase< dim > &)=default
virtual ~TensorPolynomialsBase ()=default
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

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 \(0 \le a_1, a_2, a_3 \le k\).

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.

Constructor & Destructor Documentation

◆ 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

Member Function Documentation

◆ 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

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

◆ name()

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

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)

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

◆ clone()

template<int dim>
std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsRT_Bubbles< dim >::clone ( ) const

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

Member Data Documentation

◆ raviart_thomas_space

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

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<Polynomials::Polynomial<double> > PolynomialsRT_Bubbles< dim >::monomials

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

Definition at line 145 of file polynomials_rt_bubbles.h.

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