Reference documentation for deal.II version 9.3.0
FESeries::Legendre< dim, spacedim > Class Template Reference

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

Inheritance diagram for FESeries::Legendre< dim, spacedim >:
[legend]

## Public Types

using CoefficientType = double

## Public Member Functions

Legendre (const std::vector< unsigned int > &n_coefficients_per_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection, const unsigned int component=numbers::invalid_unsigned_int)

Legendre (const unsigned int n_coefficients_per_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)

unsigned int get_n_coefficients_per_direction (const unsigned int index) const

void precalculate_all_transformation_matrices ()

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

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

bool operator== (const Legendre< dim, spacedim > &legendre) const

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

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

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

## Static Public Member Functions

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Private Attributes

const std::vector< unsigned intn_coefficients_per_direction

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

const hp::QCollection< dim > q_collection

std::vector< FullMatrix< CoefficientType > > legendre_transform_matrices

std::vector< CoefficientTypeunrolled_coefficients

const unsigned int component

## Detailed Description

### template<int dim, int spacedim = dim> class FESeries::Legendre< dim, spacedim >

A class to calculate expansion of a scalar FE (or a single component of vector-valued 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.

Definition at line 272 of file fe_series.h.

## ◆ CoefficientType

template<int dim, int spacedim = dim>
 using FESeries::Legendre< dim, spacedim >::CoefficientType = double

Definition at line 275 of file fe_series.h.

## ◆ Legendre() [1/2]

template<int dim, int spacedim>
 FESeries::Legendre< dim, spacedim >::Legendre ( const std::vector< unsigned int > & n_coefficients_per_direction, const hp::FECollection< dim, spacedim > & fe_collection, const hp::QCollection< dim > & q_collection, const unsigned int component = numbers::invalid_unsigned_int )

Constructor that initializes all required data structures.

The n_coefficients_per_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.

As the Legendre expansion can only be performed on scalar fields, this class does not operate on vector-valued finite elements and will therefore throw an assertion. However, each component of a finite element field can be treated as a scalar field, respectively, on which Legendre expansions are again possible. For this purpose, the optional parameter component defines which component of each FiniteElement will be used. The default value of component only applies to scalar FEs, in which case it indicates that the sole component is to be decomposed. For vector-valued FEs, a non-default value must be explicitly provided.

Definition at line 199 of file fe_series_legendre.cc.

## ◆ Legendre() [2/2]

template<int dim, int spacedim>
 FESeries::Legendre< dim, spacedim >::Legendre ( const unsigned int n_coefficients_per_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.

Deprecated:
Use a different constructor instead.

Definition at line 235 of file fe_series_legendre.cc.

## ◆ 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 296 of file fe_series_legendre.cc.

## ◆ get_n_coefficients_per_direction()

template<int dim, int spacedim>
 unsigned int FESeries::Legendre< dim, spacedim >::get_n_coefficients_per_direction ( const unsigned int index ) const

Return the number of coefficients in each coordinate direction for the finite element associated with index in the provided hp::FECollection.

Definition at line 285 of file fe_series_legendre.cc.

## ◆ precalculate_all_transformation_matrices()

template<int dim, int spacedim>
 void FESeries::Legendre< dim, spacedim >::precalculate_all_transformation_matrices ( )

Calculate all transformation matrices to transfer the finite element solution to the series expansion representation.

These matrices will be generated on demand by calling calculate() and stored for recurring purposes. Usually, this operation consumes a lot of workload. With this function, all matrices will be calculated in advance. This way, we can separate their costly generation from the actual application.

Definition at line 265 of file fe_series_legendre.cc.

## ◆ save_transformation_matrices()

template<int dim, int spacedim = dim>
template<class Archive >
 void FESeries::Legendre< dim, spacedim >::save_transformation_matrices ( Archive & ar, const unsigned int version )

Write all transformation matrices of this object to a stream for the purpose of serialization.

Since any of its transformation matrices has to be generated only once for a given scenario, it is common practice to determine them in advance calling precalculate_all_transformation_matrices() and keep them via serialization.

## ◆ load_transformation_matrices()

template<int dim, int spacedim = dim>
template<class Archive >
 void FESeries::Legendre< dim, spacedim >::load_transformation_matrices ( Archive & ar, const unsigned int version )

Read all transformation matrices from a stream and recover them for this object.

## ◆ operator==()

template<int dim, int spacedim>
 bool FESeries::Legendre< dim, spacedim >::operator== ( const Legendre< dim, spacedim > & legendre ) const
inline

Test for equality of two series expansion objects.

Definition at line 251 of file fe_series_legendre.cc.

## ◆ subscribe()

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

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

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

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 301 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

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

List the subscribers to the input stream.

Definition at line 318 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ serialize()

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

Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 310 of file subscriptor.h.

## ◆ n_coefficients_per_direction

template<int dim, int spacedim = dim>
 const std::vector FESeries::Legendre< dim, spacedim >::n_coefficients_per_direction
private

Number of coefficients in each direction for each finite element in the registered hp::FECollection.

Definition at line 376 of file fe_series.h.

## ◆ fe_collection

template<int dim, int spacedim = dim>
 SmartPointer > FESeries::Legendre< dim, spacedim >::fe_collection
private

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

Definition at line 381 of file fe_series.h.

## ◆ q_collection

template<int dim, int spacedim = dim>
 const hp::QCollection FESeries::Legendre< dim, spacedim >::q_collection
private

hp::QCollection used in calculation of transformation matrices.

Definition at line 386 of file fe_series.h.

## ◆ legendre_transform_matrices

template<int dim, int spacedim = dim>
 std::vector > FESeries::Legendre< dim, spacedim >::legendre_transform_matrices
private

Transformation matrices for each FiniteElement.

Definition at line 391 of file fe_series.h.

## ◆ unrolled_coefficients

template<int dim, int spacedim = dim>
 std::vector FESeries::Legendre< dim, spacedim >::unrolled_coefficients
private

Auxiliary vector to store unrolled coefficients.

Definition at line 396 of file fe_series.h.

## ◆ component

template<int dim, int spacedim = dim>
 const unsigned int FESeries::Legendre< dim, spacedim >::component
private

Which component of FiniteElement should be used to calculate the expansion.

Definition at line 402 of file fe_series.h.

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