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

Abstract

Dedicated to Professor Stanley Osher on the occasion of his 60th birthday 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. 1. Introduction. In this paper we study a class of one-sided post-processing tech- niques to enhance the accuracy of the discontinuous Galerkin methods. This is a modifi- cation of the local post-processing technique originally developed by Bramble and Schatz (1) in the context of continuous finite element methods for elliptic problems, and later by Cockburn, Luskin, Shu and Suli (6, 7) and by Ryan, Shu and Atkins (15) in the context of the discontinuous Galerkin methods for linear hyperbolic equations. Two key ingredients of this post-processing technique are a negative norm estimate for the numerical solu- tion, which should be of higher order than the L 2 error estimate, and a local translation invariance of the mesh within the support of the local post-processor. The technique then allows the recovery of the L 2 error of the post-processed solution to the order of accuracy in the negative norm estimate. The main advantages of this technique, com- pared to other post-processing techniques, include its local feature, hence its efficiency and its easiness in the parallel implementation framework, and its effectiveness in almost doubling the order of accuracy rather than increasing the order of accuracy by one or two. The original local post-processor in (1, 6, 7, 15) is based on a symmetric local stencil, using the information in about 2k neighboring elements to either side of the element being post-processed, for P k (piecewise polynomials of degree up to k) elements. The mesh must be uniform within this local stencil, and the post-processed solution is (2k+1)-th order accurate in the L 2 norm instead of the usual (k+1)-th order accuracy before post- processing, for the discontinuous Galerkin method applied to linear hyperbolic equations, which maintains a (2k+1)-th order accuracy in the negative k norm of the numerical solution. The symmetric nature of the local stencil prevents the application of the post- processor to the following situations: near a boundary of a computational domain; near a discontinuity in the solution; and near an interface of meshes with different mesh sizes. We can see clearly in the numerical examples in (15), that the post-processor fails in all