Kaskade::ScalarConverter< Cell, Scalar > Class Template Reference

A Converter for scalar shape functions that do not change their value under transformation from reference to actual element geometry. More...

#include <converter.hh>

Detailed Description

template<class Cell, class Scalar>
class Kaskade::ScalarConverter< Cell, Scalar >

A Converter for scalar shape functions that do not change their value under transformation from reference to actual element geometry.

If grid and world dimension are the same, the restriction of ansatz functions \( \varphi \) on the actual element are defined in terms of the shape functions \( \phi \) on the reference elements as

\[ \varphi(x) = \phi(B^{-1}(x-b)), \quad \xi = B^{-1}(x-b). \]

Therefore the derivatives and gradients are transformed as

\[ \varphi'(x) = \phi'(\xi)B^{-1}, \quad \nabla \varphi(x) = B^{-T}\nabla\phi(\xi). \]

Hessians (derivatives of gradients) are transformed as

\[ H\varphi(x) = B^{-T}H_\phi(\xi)B^{-1}. \]

This is used as a converter for scalar Lagrange and hierarchical spaces.

If the world dimension is larger than the grid dimension, the gradient is defined as the gradient of the function extending as a constant in normal direction. This is just

\[ \varphi'(x) = \phi'(\xi)B^{+}, \quad \nabla \varphi(x) = B^{+T}\nabla\phi(\xi), \]

where \( B \in {\bf R}^{w\times d}\) .

Todo:

: The Hessian depends on the curvature of the grid - this is not taken into account yet.

Template parameters:

Template Parameters

Cell	the type of codim 0 entity on which the shape functions are defined.

Definition at line 61 of file converter.hh.

Public Member Functions
	ScalarConverter ()
	ScalarConverter (Cell const &cell)
void	moveTo (Cell const &cell)
	Indicates that following accesses refer to the given cell. More...
void	setLocalPosition (Dune::FieldVector< typename Cell::Geometry::ctype, dim > const &xi)
Dune::FieldMatrix< Scalar, 1, 1 >	global (Dune::FieldMatrix< Scalar, 1, 1 > const &sf) const
	Applies the transformation \( \psi(x) \) to shape function value. More...
VariationalArg< Scalar, dimw, 1 >	global (VariationalArg< Scalar, dim, 1 > const &sf, int deriv=1) const
	Applies the transformation \( \psi \) to shape function value, gradient, and Hessian. More...
Dune::FieldMatrix< Scalar, 1, 1 >	local (Dune::FieldMatrix< Scalar, 1, 1 > const &glob) const

Constructor & Destructor Documentation

ScalarConverter() [1/2]

template<class Cell , class Scalar >

Kaskade::ScalarConverter< Cell, Scalar >::ScalarConverter ( )

inline

Definition at line 67 of file converter.hh.

ScalarConverter() [2/2]

template<class Cell , class Scalar >

Kaskade::ScalarConverter< Cell, Scalar >::ScalarConverter ( Cell const &  cell )

inline

Definition at line 69 of file converter.hh.

Member Function Documentation

global() [1/2]

template<class Cell , class Scalar >

Dune::FieldMatrix< Scalar, 1, 1 > Kaskade::ScalarConverter< Cell, Scalar >::global ( Dune::FieldMatrix< Scalar, 1, 1 > const &  sf ) const

inline

Applies the transformation \( \psi(x) \) to shape function value.

Definition at line 83 of file converter.hh.

Referenced by Kaskade::MorleyMapper< ScalarType, GV >::Combiner::Combiner().

global() [2/2]

template<class Cell , class Scalar >

VariationalArg< Scalar, dimw, 1 > Kaskade::ScalarConverter< Cell, Scalar >::global ( VariationalArg< Scalar, dim, 1 > const &  sf, int  deriv = 1  ) const

inline

Applies the transformation \( \psi \) to shape function value, gradient, and Hessian.

Definition at line 88 of file converter.hh.

local()

template<class Cell , class Scalar >

Dune::FieldMatrix< Scalar, 1, 1 > Kaskade::ScalarConverter< Cell, Scalar >::local ( Dune::FieldMatrix< Scalar, 1, 1 > const &  glob ) const

inline

Definition at line 97 of file converter.hh.

moveTo()

template<class Cell , class Scalar >

void Kaskade::ScalarConverter< Cell, Scalar >::moveTo ( Cell const &  cell )

inline

Indicates that following accesses refer to the given cell.

Definition at line 74 of file converter.hh.

setLocalPosition()

template<class Cell , class Scalar >

void Kaskade::ScalarConverter< Cell, Scalar >::setLocalPosition ( Dune::FieldVector< typename Cell::Geometry::ctype, dim > const &  xi )

inline

Definition at line 76 of file converter.hh.

Referenced by Kaskade::MorleyMapper< ScalarType, GV >::Combiner::Combiner().

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

• converter.hh