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

Inheritance diagram for FESeries::Legendre< dim, spacedim >:

Public Member Functions
	Legendre (const unsigned int size_in_each_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection)

template<typename Number >
void	calculate (const ::Vector< Number > &local_dof_values, const unsigned int cell_active_fe_index, Table< dim, CoefficientType > &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 (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Attributes
const unsigned int	N

SmartPointer< const hp::FECollection< dim, spacedim > >	fe_collection

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

std::vector< FullMatrix< CoefficientType > >	legendre_transform_matrices

std::vector< CoefficientType >	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, int spacedim = dim>
class FESeries::Legendre< dim, spacedim >

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 189 of file fe_series.h.

Constructor & Destructor Documentation

◆ Legendre()

template<int dim, int spacedim>

FESeries::Legendre< dim, spacedim >::Legendre	(	const unsigned int	size_in_each_direction,
		const hp::FECollection< dim, spacedim > &	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 191 of file fe_series_legendre.cc.

Member Function Documentation

◆ calculate()

template<int dim, int spacedim>

template<typename Number >

void FESeries::Legendre< dim, spacedim >::calculate	(	const ::Vector< Number > &	local_dof_values,
		const unsigned int	cell_active_fe_index,
		Table< dim, CoefficientType > &	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 207 of file fe_series_legendre.cc.