EXTENSION OF A POSTPROCESSING TECHNIQUE FOR THE DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS WITH APPLICATION TO AN AEROACOUSTIC PROBLEM ∗ JENNIFER RYAN † , CHI-WANG SHU † , AND HAROLD ATKINS ‡ SIAM J. SCI. COMPUT. c  2005 Society for Industrial and Applied Mathematics Vol. 26, No. 3, pp. 821–843 Abstract. In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94–111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577–606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions. Key words. accuracy enhancement, postprocessing, discontinuous Galerkin method, hyperbolic equations AMS subject classification. 65M60 DOI. 10.1137/S1064827503423998 1. Introduction. In this paper we further explore a local postprocessing tech- nique, originally developed by Bramble and Schatz [3] in the context of continuous finite element methods for elliptic problems, and later by Cockburn et al. [10] in the context of the discontinuous Galerkin methods for hyperbolic equations. Two key ingredients of this postprocessing technique are a negative norm estimate for the nu- merical solution, which should be of higher order than the L 2 error estimate, and a local translation invariance of the mesh. The main advantages of this technique, compared to other postprocessing techniques, include its local feature, hence its effi- ciency 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. We investigate the technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. In [10] (see also [9]), Cockburn et al. established a framework to prove the nega- tive norm estimates for discontinuous Galerkin methods applied to linear hyperbolic ∗ Received by the editors March 8, 2003; accepted for publication (in revised form) April 29, 2004; published electronically January 12, 2005. http://www.siam.org/journals/sisc/26-3/42399.html † Division of Applied Mathematics, Brown University, Providence, RI 02912 (ryanjk@ornl.gov, shu@dam.brown.edu). The research of the first author was supported by NASA Langley grant NGT- 1-01037. Current address for first author: Oak Ridge National Lab, P.O. Box 2008, MS6367, Oak Ridge, TN 37831-6367. The research of the second author was supported by ARO grant DAAD19- 00-1-0405, NSF grant DMS-0207451, and NASA Langley grant NCC1-01035. ‡ Computational Modeling and Simulations Branch, NASA Langley Research Center, Hampton, VA 23681 (h.l.atkins@larc.nasa.gov). 821