PolynomialsNedelec< dim > Class Template Reference

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

Inheritance diagram for PolynomialsNedelec< dim >:

Public Member Functions
	PolynomialsNedelec (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)

Static Private Member Functions
static std::vector< std::vector< Polynomials::Polynomial< double > > >	create_polynomials (const unsigned int k)

Private Attributes
const AnisotropicPolynomials< dim >	polynomial_space
const unsigned int	polynomial_degree
const unsigned int	n_pols

Detailed Description

template<int dim>
class PolynomialsNedelec< dim >

This class implements the first family H^curl-conforming, vector-valued polynomials, proposed by J.-C. Nédélec in 1980 (Numer. Math. 35).

The Nédélec polynomials are constructed such that the curl is in the tensor product polynomial space Q_k. Therefore, the polynomial order of each component must be one order higher in the corresponding two directions, yielding the polynomial spaces (Q_k,k+1, Q_k+1,k) and (Q_k,k+1,k+1, Q_k+1,k,k+1, Q_k+1,k+1,k) in 2d and 3d, resp.

Definition at line 50 of file polynomials_nedelec.h.

Constructor & Destructor Documentation

PolynomialsNedelec()

template<int dim>

PolynomialsNedelec< dim >::PolynomialsNedelec

(

const unsigned int

k

)

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

• k: the degree of the Nédélec space, which is the degree of the largest tensor product polynomial space Q_k contained.

Definition at line 28 of file polynomials_nedelec.cc.

Member Function Documentation

evaluate()

template<int dim>

void PolynomialsNedelec< 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 Nédélec 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 53 of file polynomials_nedelec.cc.

name()

template<int dim>

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

inlineoverridevirtual

Return the name of the space, which is Nedelec.

Implements TensorPolynomialsBase< dim >.

Definition at line 115 of file polynomials_nedelec.h.

n_polynomials()

template<int dim>

unsigned int PolynomialsNedelec< dim >::n_polynomials ( const unsigned int degree )

static

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

Definition at line 1483 of file polynomials_nedelec.cc.

clone()

template<int dim>

std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsNedelec< 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 1507 of file polynomials_nedelec.cc.

create_polynomials()

template<int dim>

std::vector< std::vector< Polynomials::Polynomial< double > > > PolynomialsNedelec< dim >::create_polynomials ( const unsigned int k )

staticprivate

A static member function that creates the polynomial space we use to initialize the polynomial_space member variable.

Definition at line 35 of file polynomials_nedelec.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>

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

inlineinherited

Return the highest polynomial degree of polynomials represented by this class. A derived class may override this if its value is different from my_degree.

Definition at line 160 of file tensor_polynomials_base.h.

Member Data Documentation

polynomial_space

template<int dim>

const AnisotropicPolynomials<dim> PolynomialsNedelec< dim >::polynomial_space

private

An object representing the polynomial space for a single component. We can re-use it by rotating the coordinates of the evaluation point.

Definition at line 102 of file polynomials_nedelec.h.

polynomial_degree

template<int dim>

const unsigned int TensorPolynomialsBase< dim >::polynomial_degree

privateinherited

The highest polynomial degree of this functions represented by this object.

Definition at line 139 of file tensor_polynomials_base.h.

n_pols

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.

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

• include/deal.II/base/polynomials_nedelec.h
• source/base/polynomials_nedelec.cc