Reference documentation for deal.II version Git cb0bd54b52 2019-09-21 16:31:22 -0400
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Public Member Functions | Static Public Member Functions | Private Attributes | List of all members
PolynomialsBDM< dim > Class Template Reference

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

Inheritance diagram for PolynomialsBDM< dim >:
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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
 
- 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 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
 

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.

Author
Guido Kanschat
Date
2003, 2005, 2009

Definition at line 101 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 Pk contained in the BDM- space.

Definition at line 30 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.

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 56 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 201 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 433 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 448 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 159 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 165 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 170 of file polynomials_bdm.h.

◆ p_values

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

Auxiliary memory.

Definition at line 175 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 180 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 185 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 190 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 195 of file polynomials_bdm.h.


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