One of the most widely used family of finite elements are the Lagrange elements, also often called Courant elements, which were first defined in :cite:`Courant43` with use of Lagrange interpolation polynomials. Their defining functionals N_{i} are given by
N_i(v) = v(\xi_i),\quad i=1,\ldots,n
where \xi_i \in\mathbb{R}^d are specific node points. As each basis function is therefore associated with a particular node they form a socalled nodal basis of P(\hat{K}).
It follows from the equations :eq:`determine-shape-fnc` and :eq:`lagrange-fnc` that
\hat{\varphi}_{i}(\xi_{j}) = \delta_{ij}
for the nodes \xi_j on the elements' domain. The number of nodes required for the definition of the shape or basis functions is determined by their polynomial order.
The Lagrange elements \mathbb{P}_1 define linear shape functions on simplicial domains. The nodes \xi_i each basis function is associated with are the vertices of the element.
.. texfigure:: fig_p1_shape_fnc.tex :align: center
The Lagrange elements \mathbb{P}_2 define quadratic shape functions and in addition to the elements' vertices use the midpoints of the elements' edges as node for their definition.
.. texfigure:: fig_p2_shape_fnc.tex :align: center