PolynomialsBDM< dim > Class Template Reference

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

Public Member Functions
	PolynomialsBDM (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 (unsigned int degree)

Private Attributes
const PolynomialSpace< dim >	polynomial_space
std::vector< Polynomials::Polynomial< double > >	monomials
unsigned int	n_pols
Threads::Mutex	mutex
std::vector< double >	p_values
std::vector< Tensor< 1, dim > >	p_grads
std::vector< Tensor< 2, dim > >	p_grad_grads
std::vector< Tensor< 3, dim > >	p_third_derivatives
std::vector< Tensor< 4, dim > >	p_fourth_derivatives

Detailed Description

template<int dim>
class PolynomialsBDM< dim >

This class implements the H^div-conforming, vector-valued Brezzi-Douglas-Marini ( BDM ) polynomials described in Brezzi and Fortin's Mixed and Hybrid Finite Element Methods (refer to pages 119 - 124).

The BDM polynomial space contain the entire \((P_{k})^{n}\) space (constructed with PolynomialSpace Legendre polynomials) as well as part of \((P_{k+1})^{n}\) (ie. \((P_{k})^{n} \subset BDM_{k} \subset (P_{k+1})^{n}\)). Furthermore, \(BDM_{k}\) elements are designed so that \(\nabla \cdot q \in P_{k-1} (K)\) and \(q \cdot n |_{e_{i}} \in P_{k}(e_{i})\). More details of two and three dimensional \(BDM_{k}\) elements are given below.

In 2D:

\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y) + r \; \text{curl} (x^{k+1}y) + s \; \text{curl} (xy^{k+1}), p_{k} \in (P_{k})^{2} \}\).

Note: the curl of a scalar function is given by \(\text{curl}(f(x,y)) = \begin{pmatrix} f_{y}(x,y) \\ -f_{x}(x,y) \end{pmatrix}\).

The basis used to construct the \(BDM_{1}\) shape functions is

\begin{align*} \phi_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \phi_1 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}x \\ 0 \end{pmatrix}, \phi_2 = \begin{pmatrix} -\sqrt{3}+2\sqrt{3}y \\ 0 \end{pmatrix}, \phi_3 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \phi_4 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}x \end{pmatrix}, \phi_5 = \begin{pmatrix} 0 \\ -\sqrt{3}+2\sqrt{3}y \end{pmatrix}, \phi_6 = \begin{pmatrix} x^2 \\ -2xy \end{pmatrix}, \phi_7 = \begin{pmatrix} 2xy \\ -y^2 \end{pmatrix}. \end{align*}

The dimension of the \(BDM_{k}\) space is \((k+1)(k+2)+2\), with \(k+1\) unknowns per edge and \(k(k-1)\) interior unknowns.

In 3D:

\( BDM_{k} = \{\mathbf{q} | \mathbf{q} = p_{k} (x,y,z) + \sum_{i=0}^{k} ( r_{i} \; \text{curl} \begin{pmatrix} 0\\0\\xy^{i+1}z^{k-i} \end{pmatrix} + s_{i} \; \text{curl} \begin{pmatrix} yz^{i+1}x^{k-i}\\0\\0 \end{pmatrix} + t_{i} \; \text{curl} \begin{pmatrix}0\\zx^{i+1}y^{k-i}\\0\end{pmatrix}) , p_{k} \in (P_{k})^{3} \}\).

Note: the 3D description of \(BDM_{k}\) is not unique. See Mixed and Hybrid Finite Element Methods page 122 for an alternative definition.

The dimension of the \(BDM_{k}\) space is \(\dfrac{(k+1)(k+2)(k+3)}{2}+3(k+1)\), with \(\dfrac{(k+1)(k+2)}{2}\) unknowns per face and \(\dfrac{(k-1)k(k+1)}{2}\) interior unknowns.

Author

Guido Kanschat

Date

2003, 2005, 2009

Definition at line 100 of file polynomials_bdm.h.

Constructor & Destructor Documentation

PolynomialsBDM()

template<int dim>

PolynomialsBDM< dim >::PolynomialsBDM

(

const unsigned int

k

)

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

• k: the degree of the BDM-space, which is the degree of the largest complete polynomial space P_k contained in the BDM- space.

Definition at line 29 of file polynomials_bdm.cc.

Member Function Documentation

compute()

template<int dim>

void PolynomialsBDM< 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

Compute the value and the first and second derivatives of each BDM 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 55 of file polynomials_bdm.cc.

n()

template<int dim>

unsigned int PolynomialsBDM< dim >::n ( ) const

inline

Return the number of BDM polynomials.

Definition at line 212 of file polynomials_bdm.h.

degree()

template<int dim>

unsigned int PolynomialsBDM< dim >::degree ( ) const

inline

Return the degree of the BDM space, which is one less than the highest polynomial degree.

Definition at line 220 of file polynomials_bdm.h.

name()

template<int dim>

std::string PolynomialsBDM< dim >::name ( ) const

inline

Return the name of the space, which is BDM.

Definition at line 228 of file polynomials_bdm.h.

compute_n_pols()

template<int dim>

unsigned int PolynomialsBDM< dim >::compute_n_pols ( unsigned int  degree )

static

Return the number of polynomials in the space BDM(degree) without requiring to build an object of PolynomialsBDM. This is required by the FiniteElement classes.

Definition at line 430 of file polynomials_bdm.cc.

Member Data Documentation

polynomial_space

template<int dim>

const PolynomialSpace<dim> PolynomialsBDM< dim >::polynomial_space

private

An object representing the polynomial space used here. The constructor fills this with the monomial basis.

Definition at line 165 of file polynomials_bdm.h.

monomials

template<int dim>

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

private

Storage for monomials. In 2D, this is just the polynomial of order k. In 3D, we need all polynomials from degree zero to k.

Definition at line 171 of file polynomials_bdm.h.

n_pols

template<int dim>

unsigned int PolynomialsBDM< dim >::n_pols

private

Number of BDM polynomials.

Definition at line 176 of file polynomials_bdm.h.

mutex

template<int dim>

Threads::Mutex PolynomialsBDM< dim >::mutex

mutableprivate

A mutex that guards the following scratch arrays.

Definition at line 181 of file polynomials_bdm.h.

p_values

template<int dim>

std::vector<double> PolynomialsBDM< dim >::p_values

mutableprivate

Auxiliary memory.

Definition at line 186 of file polynomials_bdm.h.

p_grads

template<int dim>

std::vector<Tensor<1, dim> > PolynomialsBDM< dim >::p_grads

mutableprivate

Auxiliary memory.

Definition at line 191 of file polynomials_bdm.h.

p_grad_grads

template<int dim>

std::vector<Tensor<2, dim> > PolynomialsBDM< dim >::p_grad_grads

mutableprivate

Auxiliary memory.

Definition at line 196 of file polynomials_bdm.h.

p_third_derivatives

template<int dim>

std::vector<Tensor<3, dim> > PolynomialsBDM< dim >::p_third_derivatives

mutableprivate

Auxiliary memory.

Definition at line 201 of file polynomials_bdm.h.

p_fourth_derivatives

template<int dim>

std::vector<Tensor<4, dim> > PolynomialsBDM< dim >::p_fourth_derivatives

mutableprivate

Auxiliary memory.

Definition at line 206 of file polynomials_bdm.h.

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

• deal.II/base/polynomials_bdm.h
• source/base/polynomials_bdm.cc