Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws |
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| Authors: | Juntao Huang Chi‐Wang Shu |
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| Affiliation: | 1. Zhou Pei‐Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China;2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA |
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| 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  (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree  , error estimates of  are obtained for general monotone fluxes, and optimal estimates of  are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree  , error estimates of  are obtained for general monotone fluxes, and  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 |
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| Keywords: | Discontinuous Galerkin error estimate quadrature rules conservation laws semi‐discrete |
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(piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree 
, error estimates of 
are obtained for general monotone fluxes, and optimal estimates of 
are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree 
, error estimates of 
are obtained for general monotone fluxes, and 
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