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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1402–1423 | Cite as

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

  • Waixiang Cao
  • Zhimin Zhang
Review Paper
  • 111 Downloads

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.

Keywords

Discontinuous Galerkin (DG) method LDG Direct discontinuous Galerkin (DDG) method Superconvergence Cell average 

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Authors and Affiliations

  1. 1.School of Mathematical ScienceBeijing Normal UniversityBeijingChina
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.Department of MathematicsWayne State UniversityDetroitUSA

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