Reference documentation for deal.II version Git b5bb23c 2017-06-27 10:36:53 -0400
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

std::vector< double > p_values

std::vector< Tensor< 1, dim > > p_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 Hdiv-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.

Date
2003, 2005, 2009

Definition at line 100 of file polynomials_bdm.h.

## Constructor & Destructor Documentation

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 Pk contained in the BDM- space.

Definition at line 28 of file polynomials_bdm.cc.

## Member Function Documentation

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.

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

Return the number of BDM polynomials.

Definition at line 207 of file polynomials_bdm.h.

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 215 of file polynomials_bdm.h.

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

Return the name of the space, which is BDM.

Definition at line 223 of file polynomials_bdm.h.

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 408 of file polynomials_bdm.cc.

## Member Data Documentation

template<int dim>
 const PolynomialSpace PolynomialsBDM< dim >::polynomial_space
private

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

Definition at line 160 of file polynomials_bdm.h.

template<int dim>
 std::vector > 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 166 of file polynomials_bdm.h.

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

Number of BDM polynomials.

Definition at line 171 of file polynomials_bdm.h.

template<int dim>
mutableprivate

A mutex that guards the following scratch arrays.

Definition at line 176 of file polynomials_bdm.h.

template<int dim>
 std::vector PolynomialsBDM< dim >::p_values
mutableprivate

Auxiliary memory.

Definition at line 181 of file polynomials_bdm.h.

template<int dim>
mutableprivate

Auxiliary memory.

Definition at line 186 of file polynomials_bdm.h.

template<int dim>
mutableprivate

Auxiliary memory.

Definition at line 191 of file polynomials_bdm.h.

template<int dim>
 std::vector > PolynomialsBDM< dim >::p_third_derivatives
mutableprivate

Auxiliary memory.

Definition at line 196 of file polynomials_bdm.h.

template<int dim>
 std::vector > PolynomialsBDM< dim >::p_fourth_derivatives
mutableprivate

Auxiliary memory.

Definition at line 201 of file polynomials_bdm.h.

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