Analysis of Optimal Error Estimates and Superconvergence of the Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension
Year: 2016
International Journal of Numerical Analysis and Modeling, Vol. 13 (2016), Iss. 3 : pp. 403–434
Abstract
In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ convergence rates in the $L^2$-norm, respectively, when $p$-degree piecewise polynomials with $p\geqslant 1$ are used. We further prove that the $p$-degree DG solution and its derivative are $\mathcal{O}(h^{2p})$ superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2016-IJNAM-446
International Journal of Numerical Analysis and Modeling, Vol. 13 (2016), Iss. 3 : pp. 403–434
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Discontinuous Galerkin method convection-diffusion problems singularly pertur- bed problems superconvergence upwind and downwind points Shishkin meshes.