Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k + 1/2 in the L²-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L² only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L² (Ω₀), where Ω₀ is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.