Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions

Authors:
Liangyue Ji, Yan Xu and Jennifer K. Ryan

Journal:
Math. Comp. **81** (2012), 1929-1950

MSC (2010):
Primary 65M60; Secondary 35K10, 35L02

DOI:
https://doi.org/10.1090/S0025-5718-2012-02586-5

Published electronically:
March 2, 2012

MathSciNet review:
2945143

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Abstract | References | Similar Articles | Additional Information

Abstract: Discontinuous Galerkin (DG) methods exhibit ``hidden accuracy'' that makes superconvergence of this method an increasing popular topic to address. Previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations. Additionally, superconvergence of order for the convection-diffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Shu. In this paper we demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for use on the multi-dimensional linear convection-diffusion equation in order to obtain 2+ order of accuracy, where depends upon the flux and takes on the values or The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving filter. We solve this convection-diffusion equation using the local discontinuous Galerkin (LDG) method and show theoretically that it is possible to obtain in the negative-order norm. By post-processing the LDG solution to a linear convection equation using a specially designed kernel such as the one by Cockburn et al., we can compute this same order accuracy in the -norm. Additionally, we present numerical studies that confirm that we can improve the LDG solution from to using alternating fluxes and that we actually obtain for diffusion-dominated problems.

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Additional Information

**Liangyue Ji**

Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands.

Address at time of publication:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.

Email:
jlyue@mail.ustc.edu.cn

**Yan Xu**

Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

Email:
yxu@ustc.edu.cn

**Jennifer K. Ryan**

Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands

Email:
J.K.Ryan@tudelft.nl

DOI:
https://doi.org/10.1090/S0025-5718-2012-02586-5

Keywords:
discontinuous Galerkin method,
convection-diffusion equations,
negative-order norm error estimates,
filtering,
post-processing,
accuracy enhancement.

Received by editor(s):
September 26, 2010

Received by editor(s) in revised form:
April 26, 2011, and July 1, 2011

Published electronically:
March 2, 2012

Additional Notes:
The research of the second author was supported by NSFC grant No.10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 09-0922 and the Fundamental Research Funds for the Central Universities. Additional support was provided by the Alexander von Humboldt-Foundation while the author was in residence at Freiburg University, Germany

Article copyright:
© Copyright 2012
American Mathematical Society