The Reference ElementΒΆ

In the sense of Ciarlet [Cia78] a finite element is defined as a triple \((\hat{K}, P(\hat{K}), \Sigma(\hat{K}))\) where \(\hat{K} \subset \mathbb{R}^d\) is a closed bounded subset with non-empty interior, \(P(\hat{K})\) is a finite dimensional vector space on \(\hat{K}\) and \(\Sigma(\hat{K})\) is a basis of the dual space \(P'(\hat{K})\), i.e. the space of linear functionals on \(P(\hat{K})\).

In most cases \(\hat{K}\) is a polygonial domain and \(P(\hat{K})\) is a polynomial function space. The elements \(\hat{\varphi}_j, j=1,\ldots,n\) of a basis of \(P(\hat{K})\), with \(n\in\mathbb{N}\) being its dimension, are called shape functions and the linear functionals \(N_i \in\Sigma(\hat{K}), i=1,\ldots,n\) define the socalled degrees of freedom.

The shape functions \(\hat{\varphi}_j\) are determined by the linear functionals \(N_i\) via

\[N_{i}(\hat{\varphi}_{j}) = \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & \text{otherwise} \end{cases}\]

so the choice of these functionals provides different families of finite elements (see [LB13]).

The purpose of the reference element is to provide methods for the evaluation of these shape or basis functions \(\hat{\varphi}\) defined on the reference domain and their derivatives.