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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

   

 

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
Published electronically: March 2, 2012
MathSciNet review: 2945143
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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 $ k+\frac {3}{2}$ 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$ k$+$ m$ order of accuracy, where $ m$ depends upon the flux and takes on the values $ 0,\, \frac {1}{2},$ or $ 1.$ 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 $ \mathcal {O}(h^{2k+m})$ 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 $ L^2$-norm. Additionally, we present numerical studies that confirm that we can improve the LDG solution from $ \mathcal {O}(h^{k+1})$ to $ \mathcal {O}(h^{2k+1})$ using alternating fluxes and that we actually obtain $ \mathcal {O}(h^{2k+2})$ 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: http://dx.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