Enhanced Accuracy by Post-processing for Finite Element Methods for Hyperbolic Equations

摘要

Abstract. We consider the enhancement of accuracy, by means of a simple post-processing technique, for nite 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 the mesh size 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 L 2 norm, whereas the post-processed approximation is of order 2k +1; if the exact solution is in L 2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1=2 inL 2 over a subdomain on which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.