Reference documentation for deal.II version 9.3.0
PolynomialsNedelec< dim > Class Template Reference

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

Inheritance diagram for PolynomialsNedelec< dim >:
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## 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

## Detailed Description

### template<int dim> class PolynomialsNedelec< dim >

This class implements the first family Hcurl-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 Qk. Therefore, the polynomial order of each component must be one order higher in the corresponding two directions, yielding the polynomial spaces (Qk,k+1, Qk+1,k) and (Qk,k+1,k+1, Qk+1,k,k+1, Qk+1,k+1,k) in 2D and 3D, resp.

Definition at line 51 of file polynomials_nedelec.h.

## ◆ 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 Qk contained.

Definition at line 29 of file polynomials_nedelec.cc.

## ◆ 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.

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 54 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 121 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 1489 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 1513 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 36 of file polynomials_nedelec.cc.

## ◆ n()

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

Return the number of polynomials.

Definition at line 155 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 164 of file tensor_polynomials_base.h.

## ◆ polynomial_space

template<int dim>
 const AnisotropicPolynomials 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 108 of file polynomials_nedelec.h.

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