On a One-Sided Post-Processing Technique for the Discontinuous Galerkin Methods

Abstract

In this paper we study a class of one-sided post-processing techniques to enhance the accuracy of the discontinuous Galerkin methods. The applications considered in this paper are linear hyperbolic equations, however the technique can be used for the solution to a discontinuous Galerkin method solving other types of partial differential equations, or more general approximations, as long as there is a higher order negative norm error estimate for the numerical solution. The advantage of the one-sided post-processing is that it uses information only from one side, hence it can be applied up to domain boundaries, a discontinuity in the solution, or an interface of different mesh sizes. This technique allows us to obtain an improvement in the order of accuracy from k+1 of the discontinuous Galerkin method to 2k+1 of the post-processed solution, using piecewise polynomials of degree k, throughout the entire domain and not just away from the boundaries, discontinuities, or interfaces of different mesh sizes.