Lagrange Elements¶
One of the most widely used family of finite elements are the Lagrange elements, also often called Courant elements, which were first defined in [Cou43] with use of Lagrange interpolation polynomials. Their defining functionals \(N_{i}\) are given by
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 (?) and (1) that
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.
\(\mathbb{P}_1\) Elements¶
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.
\(\mathbb{P}_2\) Elements¶
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.