Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations |
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| Authors: | Waixiang Cao Hailiang Liu Zhimin Zhang |
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| Institution: | 1. Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing, China;2. Department of Mathematics, Iowa State University, Ames, Iowa;3. Department of Mathematics, Wayne State University, Detroit, Michigan |
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| Abstract: | This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and  ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a  ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017 |
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| Keywords: | convection‐diffusion equations direct discontinuous Galerkin methods superconvergence |
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‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a 
‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017