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Superconvergence of Local Discontinuous Galerkin methods for one-dimensional linear parabolic equations


Authors: Waixiang Cao and Zhimin Zhang
Journal: Math. Comp. 85 (2016), 63-84
MSC (2010): Primary 65M15, 65M60, 65N30
Published electronically: June 1, 2015
MathSciNet review: 3404443
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Abstract: In this paper, we study superconvergence properties of the local discontinuous Galerkin method for one-dimensional linear parabolic equations when alternating fluxes are used. We prove, for any polynomial degree $ k$, that the numerical fluxes converge at a rate of $ 2k+1$ (or $ 2k+1/2$) for all mesh nodes and the domain average under some suitable initial discretization. We further prove a $ k+1$th superconvergence rate for the derivative approximation and a $ k+2$th superconvergence rate for the function value approximation at the Radau points. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp.


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Additional Information

Waixiang Cao
Affiliation: Beijing Computational Science Research Center, Beijing, 100084, China — and — College of Mathematics and Computational Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing, 100084, China — and — Department of Mathematics, Wayne State University, Detroit, Michigan 48202

DOI: https://doi.org/10.1090/mcom/2975
Keywords: Local discontinuous Galerkin method, superconvergence, parabolic, Radau points, cell average, initial discretization
Received by editor(s): February 12, 2014
Received by editor(s) in revised form: May 10, 2014
Published electronically: June 1, 2015
Additional Notes: The second author was supported in part by the National Natural Science Foundation of China (NSFC) under grant Nos. 11471031, 91430216, and the US National Science Foundation (NSF) through grant DMS-1419040
Article copyright: © Copyright 2015 American Mathematical Society