Extending linear finite elements to quadratic precision on arbitrary meshes

The method of finite elements can provide discrete versions of differential operators in general arrangements of nodes. Linear finite elements defined on a triangulation, like Delaunay's, constitute a typical route towards this aim, as well as being a tool for interpolation. We discuss a procedure to build interpolating functions with quadratic precision from this functional set. The idea is to incorporate (extend) the products of the original functions in order to go from linear precision (the property that linear functions are reconstructed exactly) to quadratic precision (quadratic functions are reconstructed exactly). This procedure is applied to the standard Galerkin approach, in order to study the Poisson problem and the direct evaluation of the Laplacian. We discuss convergence of these problems in 1D and 2D. The extended method is shown to be superior in 2D, featuring convergence for the direct evaluation of the Laplacian in distorted lattices.