In the sense of Ciarlet :cite:`Ciarlet78` 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 :cite:`Larson13`).
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.
.. toctree:: :maxdepth: 2 lagrange