Superconvergence of Local Discontinuous Galerkin methods for one-dimensional linear parabolic equations
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- by Waixiang Cao and Zhimin Zhang PDF
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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.References
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Additional Information
- Waixiang Cao
- Affiliation: Beijing Computational Science Research Center, Beijing, 100084, China β and β 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 β and β Department of Mathematics, Wayne State University, Detroit, Michigan 48202
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 63-84
- MSC (2010): Primary 65M15, 65M60, 65N30
- DOI: https://doi.org/10.1090/mcom/2975
- MathSciNet review: 3404443