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

Inheritance diagram for PolynomialsBDM< dim >:

Public Member Functions
	PolynomialsBDM (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 PolynomialSpace< dim >	polynomial_space

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

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

const unsigned int	polynomial_degree

const unsigned int	n_pols

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.

Definition at line 98 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

◆ evaluate()

template<int dim>

void PolynomialsBDM< 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

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.

Implements TensorPolynomialsBase< dim >.

Definition at line 55 of file polynomials_bdm.cc.

◆ name()

template<int dim>

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

inlineoverridevirtual

Return the name of the space, which is BDM.

Implements TensorPolynomialsBase< dim >.

Definition at line 193 of file polynomials_bdm.h.

◆ n_polynomials()

template<int dim>

unsigned int PolynomialsBDM< dim >::n_polynomials ( const 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 361 of file polynomials_bdm.cc.

◆ clone()

template<int dim>

std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsBDM< dim >::clone ( ) const

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 376 of file polynomials_bdm.cc.

◆ n()

template<int dim>

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

inlineinherited

Return the number of polynomials.

Definition at line 151 of file tensor_polynomials_base.h.

◆ degree()

template<int dim>

template<int dim>

const unsigned int TensorPolynomialsBase< dim >::n_pols

privateinherited

The number of polynomials represented by this object.

Definition at line 144 of file tensor_polynomials_base.h.

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

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

Public Member Functions

Static Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ PolynomialsBDM()

Member Function Documentation

◆ evaluate()

◆ name()

◆ n_polynomials()

◆ clone()

◆ n()

◆ degree()

Member Data Documentation

◆ polynomial_space

◆ monomials

◆ mutex

◆ p_values

◆ p_grads

◆ p_grad_grads

◆ p_third_derivatives

◆ p_fourth_derivatives

◆ polynomial_degree

◆ n_pols