FESeries::Legendre< dim > Class Template Reference

#include <deal.II/fe/fe_series.h>

Inheritance diagram for FESeries::Legendre< dim >:

Public Member Functions
	Legendre (const unsigned int size_in_each_direction, const hp::FECollection< dim > &fe_collection, const hp::QCollection< dim > &q_collection)
void	calculate (const ::Vector< double > &local_dof_values, const unsigned int cell_active_fe_index, Table< dim, double > &legendre_coefficients)
Public Member Functions inherited from Subscriptor
	Subscriptor ()
	Subscriptor (const Subscriptor &)
	Subscriptor (Subscriptor &&) noexcept
virtual	~Subscriptor ()
Subscriptor &	operator= (const Subscriptor &)
Subscriptor &	operator= (Subscriptor &&) noexcept
void	subscribe (const char *identifier=nullptr) const
void	unsubscribe (const char *identifier=nullptr) const
unsigned int	n_subscriptions () const
void	list_subscribers () const
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Member Functions
void	ensure_existence (const unsigned int fe_index)

Private Attributes
const unsigned int	N
SmartPointer< const hp::FECollection< dim > >	fe_collection
SmartPointer< const hp::QCollection< dim > >	q_collection
std::vector< FullMatrix< double > >	legendre_transform_matrices
std::vector< double >	unrolled_coefficients

Additional Inherited Members
Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)
static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Detailed Description

template<int dim>
class FESeries::Legendre< dim >

A class to calculate expansion of a scalar FE field into series of Legendre functions on a reference element.

Legendre functions are solutions to Legendre's differential equation

\[ \frac{d}{dx}\left([1-x^2] \frac{d}{dx} P_n(x)\right) + n[n+1] P_n(x) = 0 \]

and can be expressed using Rodrigues' formula

\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[x^2-1]^n. \]

These polynomials are orthogonal with respect to the \( L^2 \) inner product on the interval \( [-1;1] \)

\[ \int_{-1}^1 P_m(x) P_n(x) = \frac{2}{2n + 1} \delta_{mn} \]

and are complete. A family of \( L^2 \)-orthogonal polynomials on \( [0;1] \) can be constructed via

\[ \widetilde P_m = \sqrt{2} P_m(2x-1). \]

An arbitrary scalar FE field on the reference element \( [0;1] \) can be expanded in the complete orthogonal basis as

\[ u(x) = \sum_{m} c_m \widetilde P_{m}(x). \]

From the orthogonality property of the basis, it follows that

\[ c_m = \frac{2m+1}{2} \int_0^1 u(x) \widetilde P_m(x) dx . \]

This class calculates coefficients \( c_{\bf k} \) using \( dim \)-dimensional Legendre polynomials constructed from \( \widetilde P_m(x) \) using tensor product rule.

Author

Denis Davydov, 2016.

Definition at line 188 of file fe_series.h.

Constructor & Destructor Documentation

Legendre()

template<int dim>

FESeries::Legendre< dim >::Legendre	(	const unsigned int	size_in_each_direction,
		const hp::FECollection< dim > &	fe_collection,
		const hp::QCollection< dim > &	q_collection
	)

A non-default constructor. The size_in_each_direction defines the number of coefficients in each direction, fe_collection is the hp::FECollection for which expansion will be used and q_collection is the hp::QCollection used to integrate the expansion for each FiniteElement in fe_collection.

Definition at line 245 of file fe_series.cc.

Member Function Documentation

calculate()

template<int dim>

void FESeries::Legendre< dim >::calculate	(	const ::Vector< double > &	local_dof_values,
		const unsigned int	cell_active_fe_index,
		Table< dim, double > &	legendre_coefficients
	)

Calculate legendre_coefficients of the cell vector field given by local_dof_values corresponding to FiniteElement with cell_active_fe_index .

Definition at line 259 of file fe_series.cc.

ensure_existence()

template<int dim>

void FESeries::Legendre< dim >::ensure_existence ( const unsigned int  fe_index )

private

Ensure that the transformation matrix for FiniteElement index fe_index is calculated. If not, calculate it.

Member Data Documentation

N

template<int dim>

const unsigned int FESeries::Legendre< dim >::N

private

Number of coefficients in each direction

Definition at line 215 of file fe_series.h.

fe_collection

template<int dim>

SmartPointer<const hp::FECollection<dim> > FESeries::Legendre< dim >::fe_collection

private

hp::FECollection for which transformation matrices will be calculated.

Definition at line 220 of file fe_series.h.

q_collection

template<int dim>

SmartPointer<const hp::QCollection<dim> > FESeries::Legendre< dim >::q_collection

private

hp::QCollection used in calculation of transformation matrices.

Definition at line 225 of file fe_series.h.

legendre_transform_matrices

template<int dim>

std::vector<FullMatrix<double> > FESeries::Legendre< dim >::legendre_transform_matrices

private

Transformation matrices for each FiniteElement.

Definition at line 236 of file fe_series.h.

unrolled_coefficients

template<int dim>

std::vector<double> FESeries::Legendre< dim >::unrolled_coefficients

private

Auxiliary vector to store unrolled coefficients.

Definition at line 241 of file fe_series.h.

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

• deal.II/fe/fe_series.h
• source/fe/fe_series.cc