PolynomialSpace< dim > Class Template Reference

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

Inheritance diagram for PolynomialSpace< dim >:

Public Member Functions
template<class Pol >
	PolynomialSpace (const std::vector< Pol > &pols)
template<class StreamType >
void	output_indices (StreamType &out) const
void	set_numbering (const std::vector< unsigned int > &renumber)
void	compute (const Point< dim > &unit_point, std::vector< double > &values, std::vector< Tensor< 1, dim >> &grads, std::vector< Tensor< 2, dim >> &grad_grads, std::vector< Tensor< 3, dim >> &third_derivatives, std::vector< Tensor< 4, dim >> &fourth_derivatives) const
double	compute_value (const unsigned int i, const Point< dim > &p) const
template<int order>
Tensor< order, dim >	compute_derivative (const unsigned int i, const Point< dim > &p) const
Tensor< 1, dim >	compute_grad (const unsigned int i, const Point< dim > &p) const
Tensor< 2, dim >	compute_grad_grad (const unsigned int i, const Point< dim > &p) const
unsigned int	n () const
unsigned int	degree () const

Static Public Member Functions
static unsigned int	compute_n_pols (const unsigned int n)

Static Public Attributes
static const unsigned int	dimension = dim

Protected Member Functions
std::array< unsigned int, dim >	compute_index (const unsigned int n) const

Private Attributes
const std::vector< Polynomials::Polynomial< double > >	polynomials
const unsigned int	n_pols
std::vector< unsigned int >	index_map
std::vector< unsigned int >	index_map_inverse

Detailed Description

template<int dim>
class PolynomialSpace< dim >

Representation of the space of polynomials of degree at most n in higher dimensions.

Given a vector of n one-dimensional polynomials P₀ to P_n, where P_i has degree i, this class generates all dim-dimensional polynomials of the form P_ijk(x,y,z) = P_i(x)P_j(y)P_k(z), where the sum of i, j and k is less than or equal n.

The output_indices() function prints the ordering of the polynomials, i.e. for each dim-dimensional polynomial in the polynomial space it gives the indices i,j,k of the one-dimensional polynomials in x,y and z direction. The ordering of the dim-dimensional polynomials can be changed by using the set_numbering() function.

The standard ordering of polynomials is that indices for the first space dimension vary fastest and the last space dimension is slowest. In particular, if we take for simplicity the vector of monomials x⁰, x¹, x²,..., xⁿ, we get

1D

x⁰, x¹,...,xⁿ

2D:

x⁰y⁰, x¹y⁰,..., xⁿy⁰,
x⁰y¹, x¹y¹,..., x^n-1y¹,
x⁰y²,... x^n-2y²,
...
x⁰y^n-1, x¹y^n-1,
x⁰yⁿ

3D:

x⁰y⁰z⁰,..., xⁿy⁰z⁰,
x⁰y¹z⁰,..., x^n-1y¹z⁰,
...
x⁰yⁿz⁰,
x⁰y⁰z¹,... x^n-1y⁰z¹,
...
x⁰y^n-1z¹,
x⁰y⁰z²,... x^n-2y⁰z²,
...
x⁰y⁰zⁿ

Author

Guido Kanschat, Wolfgang Bangerth, Ralf Hartmann 2002, 2003, 2004, 2005

Definition at line 98 of file polynomial_space.h.

Constructor & Destructor Documentation

PolynomialSpace()

template<int dim>

template<class Pol >

PolynomialSpace< dim >::PolynomialSpace

(

const std::vector< Pol > &

pols

)

Constructor. pols is a vector of pointers to one-dimensional polynomials and will be copied into a private member variable. The static type of the template argument pols needs to be convertible to Polynomials::Polynomial<double>, i.e. should usually be a derived class of Polynomials::Polynomial<double>.

Definition at line 272 of file polynomial_space.h.

Member Function Documentation

output_indices()

template<int dim>

template<class StreamType >

void PolynomialSpace< dim >::output_indices

(

StreamType &

out

)

const

Prints the list of the indices to out.

Definition at line 310 of file polynomial_space.h.

set_numbering()

template<int dim>

void PolynomialSpace< dim >::set_numbering

(

const std::vector< unsigned int > &

renumber

)

Set the ordering of the polynomials. Requires renumber.size()==n(). Stores a copy of renumber.

Definition at line 112 of file polynomial_space.cc.

compute()

template<int dim>

void PolynomialSpace< dim >::compute	(	const Point< dim > &	unit_point,
		std::vector< double > &	values,
		std::vector< Tensor< 1, dim >> &	grads,
		std::vector< Tensor< 2, dim >> &	grad_grads,
		std::vector< Tensor< 3, dim >> &	third_derivatives,
		std::vector< Tensor< 4, dim >> &	fourth_derivatives
	)		const

Compute the value and the first and second derivatives of each polynomial at unit_point.

The size of the vectors must either be equal 0 or equal n(). In the first case, the function will not compute these values, i.e. you indicate what you want to have computed by resizing those vectors which you want filled.

If you need values or derivatives of all 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 polynomials.

Definition at line 203 of file polynomial_space.cc.

compute_value()

template<int dim>

double PolynomialSpace< dim >::compute_value	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

Compute the value of the ith polynomial at unit point p.

Consider using compute() instead.

Definition at line 126 of file polynomial_space.cc.

compute_derivative()

template<int dim>

template<int order>

Tensor< order, dim > PolynomialSpace< dim >::compute_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

Compute the orderth derivative of the ith polynomial at unit point p.

Consider using compute() instead.

Template Parameters

order

The order of the derivative.

Definition at line 325 of file polynomial_space.h.

compute_grad()

template<int dim>

Tensor< 1, dim > PolynomialSpace< dim >::compute_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

Compute the gradient of the ith polynomial at unit point p.

Consider using compute() instead.

Definition at line 143 of file polynomial_space.cc.

compute_grad_grad()

template<int dim>

Tensor< 2, dim > PolynomialSpace< dim >::compute_grad_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

Compute the second derivative (grad_grad) of the ith polynomial at unit point p.

Consider using compute() instead.

Definition at line 168 of file polynomial_space.cc.

n()

template<int dim>

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

inline

Return the number of polynomials spanning the space represented by this class. Here, if N is the number of one-dimensional polynomials given, then the result of this function is N in 1d, N(N+1)/2 in 2d, and N(N+1)(N+2)/6 in 3d.

Definition at line 292 of file polynomial_space.h.

degree()

template<int dim>

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

inline

Degree of the space. This is by definition the number of polynomials given to the constructor, NOT the maximal degree of a polynomial in this vector. The latter value is never checked and therefore left to the application.

Definition at line 301 of file polynomial_space.h.

compute_n_pols()

template<int dim>

unsigned int PolynomialSpace< dim >::compute_n_pols ( const unsigned int  n )

static

Static function used in the constructor to compute the number of polynomials.

Warning

The argument n is not the maximal degree, but the number of onedimensional polynomials, thus the degree plus one.

Definition at line 25 of file polynomial_space.cc.

compute_index()

template<int dim>

std::array<unsigned int, dim> PolynomialSpace< dim >::compute_index ( const unsigned int  n ) const

protected

Compute numbers in x, y and z direction. Given an index n in the d-dimensional polynomial space, return the indices i,j,k such that p_n(x,y,z) = p_i(x)p_j(y)p_k(z).

In 1d and 2d, obviously only i and i,j are returned.

Member Data Documentation

dimension

template<int dim>

const unsigned int PolynomialSpace< dim >::dimension = dim

static

Access to the dimension of this object, for checking and automatic setting of dimension in other classes.

Definition at line 105 of file polynomial_space.h.

polynomials

template<int dim>

const std::vector<Polynomials::Polynomial<double> > PolynomialSpace< dim >::polynomials

private

Copy of the vector pols of polynomials given to the constructor.

Definition at line 235 of file polynomial_space.h.

n_pols

template<int dim>

const unsigned int PolynomialSpace< dim >::n_pols

private

Store the precomputed value which the n() function returns.

Definition at line 240 of file polynomial_space.h.

index_map

template<int dim>

std::vector<unsigned int> PolynomialSpace< dim >::index_map

private

Index map for reordering the polynomials.

Definition at line 245 of file polynomial_space.h.

index_map_inverse

template<int dim>

std::vector<unsigned int> PolynomialSpace< dim >::index_map_inverse

private

Index map for reordering the polynomials.

Definition at line 250 of file polynomial_space.h.

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

• deal.II/base/polynomial_space.h
• source/base/polynomial_space.cc