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Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations


Author: Hailiang Liu
Journal: Math. Comp. 84 (2015), 2263-2295
MSC (2010): Primary 35K15, 65M15, 65M60, 76R50
DOI: https://doi.org/10.1090/S0025-5718-2015-02923-8
Published electronically: February 17, 2015
MathSciNet review: 3356026
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Abstract: In this paper, we present the optimal $ L^2$-error estimate of $ O(h^{k+1})$ for polynomial elements of degree $ k$ of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of $ k+1$ hinge on a superconvergence estimate when tensor product polynomials of degree $ k$ are used on Cartesian grids. A collection of projection errors in both one- and multi-dimensional cases is established.


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Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50010
Email: hliu@iastate.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02923-8
Keywords: Convection-diffusion equations, discontinuous Galerkin, global projection, numerical flux
Received by editor(s): February 19, 2013
Received by editor(s) in revised form: August 13, 2013, and December 12, 2013
Published electronically: February 17, 2015
Article copyright: © Copyright 2015 American Mathematical Society