provides Local Finite Elements. More...

Classes
class	ug::LFEID
	Identifier for Local Finite Elements. More...

class	ug::LocalDoF

class	ug::LocalShapeFunctionSet< TDim, TShape, TGrad >
	virtual base class for local shape function sets More...

Detailed Description

provides Local Finite Elements.

The Local Finite Element section is used to describe finite element spaces by their definition on reference elements.

A Finite Element is defined as a triplet \( \{ K, P, \Sigma \} \) (See e.g. Ciarlet, P., "Basis Error Estimates for Elliptic Problems", North-Holland, Amsterdam, 1991, p. 93; or Ern, A. and Guermond J.L., "Theory and Practice of Finite Elements", Springer, 2004, p. 19), where

\( K \) is a compact, connected, Lipschitz subset of \(\mathbb{R}^d\) with non-empty interior
\( P \) is a vector space of functions \(p: K \mapsto \mathbb{R}^m \) with an integer \(m > 0 \) (usually \(m=1\) or \(m=d\))
\(\Sigma\) is a set of \( n_{sh} \) linear forms \( \sigma_1, \dots, \sigma_{n_{sh}} \) acting on the elements of \( P \), such that the linear mapping
\[ p \mapsto ( \sigma_1(p), \dots, \sigma_{n_{sh}}(p)) \in \mathbb{R}^{n_{sh}} \]
is bijective. These linear forms are called local degrees of freedom.

Since the mapping is bijective, there exist a basis \(\{\phi_1, \dots, \phi_{n_{sh}} \subset P\) such that

\[ \sigma_i (\phi_j) = \delta_{ij}, \qquad 1 \leq i,j \leq n_{sh}. \]

This set is called the set of local shape functions. The implemented counterpart is the class LocalShapeFunctionSet.

The set of local degrees of freedom finds its counterpart in the class ILocalDoFSet.

Classes

Detailed Description