#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 &&)

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&)

void	subscribe (const char *identifier=0) const

void	unsubscribe (const char *identifier=0) 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, char *arg2, std::string &arg3)

static ::ExceptionBase &	ExcNoSubscriber (char arg1, char 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.

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

deal.II/fe/fe_series.h
/build/deal.ii-IXg7Js/deal.ii-8.5.1/source/fe/fe_series.cc

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

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

Constructor & Destructor Documentation

◆ Legendre()

Member Function Documentation

◆ calculate()

◆ ensure_existence()

Member Data Documentation

◆ N

◆ fe_collection

◆ q_collection

◆ legendre_transform_matrices

◆ unrolled_coefficients

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