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Some Recent Developments in Superconvergence of Discontinuous Galerkin Methods for Time-Dependent Partial Differential Equations

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Abstract

In this paper, we briefly review some recent developments in the superconvergence of three types of discontinuous Galerkin (DG) methods for time-dependent partial differential equations: the standard DG method, the local discontinuous Galerkin method, and the direct discontinuous Galerkin method. A survey of our own results for various time-dependent partial differential equations is presented and the superconvergence phenomena of the aforementioned three types of DG solutions are studied for: (i) the function value and derivative approximation at some special points, (ii) cell average error and supercloseness.

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Authors and Affiliations

  1. School of Mathematical Science, Beijing Normal University, Beijing, China

    Waixiang Cao

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Waixiang Cao
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Correspondence to Waixiang Cao.

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Waixiang Cao: Research supported in part by NSFC Grant No. 11501026, and and the Fundamental Research Funds for the Central Universities No. 2017NT10. Zhimin Zhang: Research supported in part by the following grants: NSFC 1147103 and 91430216, NSAF1530401, and NSF DMS-1419040.

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Cao, W., Zhang, Z. Some Recent Developments in Superconvergence of Discontinuous Galerkin Methods for Time-Dependent Partial Differential Equations. J Sci Comput 77, 1402–1423 (2018). https://doi.org/10.1007/s10915-018-0762-2

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