Post-Processing of Galerkin Methods for Hyperbolic Problems

It is well known that the discontinuous Galerkin (DG) method for scalar linear conservation laws has an order of convergence of k + 1/2 when polynomials of degree k are used and the exact solution is sufficiently smooth. In this paper, we show that a suitable post-processing of the DG approximate solution is of order 2k+1 in L2(Ω₀) where Ω₀ is a domain on which the exact solution is smooth enough. The post-processing is a convolution with a kernel whose support has measure of order one if the meshes are arbitrary; if the meshes are translation invariant, the support of the kernel is a cube whose edges are of size of order ∆x only. The post-processing has to be performed only once, at the final time level.