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

Abstract:

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 $\Delta x$ 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 $k+1/2$ in $L^2(\Omega _0)$, where $\Omega _0$ is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.