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Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws
Authors:Juntao Huang  Chi‐Wang Shu
Institution:1. Zhou Pei‐Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China;2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA
Abstract:In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with urn:x-wiley:0749159X:media:num22089:num22089-math-0001 (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree urn:x-wiley:0749159X:media:num22089:num22089-math-0002, error estimates of urn:x-wiley:0749159X:media:num22089:num22089-math-0003 are obtained for general monotone fluxes, and optimal estimates of urn:x-wiley:0749159X:media:num22089:num22089-math-0004 are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree urn:x-wiley:0749159X:media:num22089:num22089-math-0005, error estimates of urn:x-wiley:0749159X:media:num22089:num22089-math-0006 are obtained for general monotone fluxes, and urn:x-wiley:0749159X:media:num22089:num22089-math-0007 are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017
Keywords:Discontinuous Galerkin  error estimate  quadrature rules  conservation laws  semi‐discrete
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