Postprocessing for the Discontinuous Galerkin Method over Nonuniform Meshes

A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Süli [Proceedings of the International Symposium on Discontinuous Galerkin Methods, Springer, New York, pp. 291-300; Math. Comput., 72 (2003), pp. 577-606]. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order $k$+1 to order 2$k$+1 over a uniform mesh. Assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix-vector multiplications. In this paper, we present two alternatives for extending this postprocessing technique to include smoothly varying meshes. The first method uses a simple local $L^2$-projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution. By using this local $L^2$-projection, recalculating the convolution kernel for every element can be avoided, and 2$k$+1 order accuracy of the postprocessed solution can be achieved. The second method uses the idea of characteristic length based upon the largest element size for the scaling of the postprocessing kernel. These two methods, local projection and characteristic length, are also applied to approximations over a mesh with elements that vary in size randomly. We discuss the computational issues in using these two techniques and demonstrate numerically that we obtain the 2$k$+1 order of accuracy for the smoothly varying meshes, and that although the 2$k$+1 order of accuracy is not fully realized for random meshes, there is significant improvement in the $L^2$-errors.