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Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D

Authors: Ziqing Xie and Zhimin Zhang
Journal: Math. Comp. 79 (2010), 35-45
MSC (2000): Primary 65N30, 65N15
Published electronically: August 3, 2009
MathSciNet review: 2552216
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Abstract: It has been observed from the authors' numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed two-point boundary problems of the convection-diffusion type. Especially when using a piecewise polynomial space of degree $ k$, the LDG solution achieves the optimal convergence rate $ k+1$ under the $ L^2$-norm, and a superconvergence rate $ 2k+1$ for the one-sided flux uniformly with respect to the singular perturbation parameter $ \epsilon$. In this paper, we investigate the theoretical aspect of this phenomenon under a simplified ODE model. In particular, we establish uniform convergence rates $ \sqrt{\epsilon} \left( \frac{\ln N}{N} \right)^{k+1}$ for the $ L^2$-norm and $ \left (\frac{\ln N}{N} \right)^{2k+1}$ for the one-sided flux inside the boundary layer region. Here $ N$ (even) is the number of elements.

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Additional Information

Ziqing Xie
Affiliation: College of Mathematics and Computer Science, Hunan Normal University, People’s Republic of China

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Superconvergence, discontinuous Galerkin method, finite element, singularly perturbed problem, convection-diffusion.
Received by editor(s): March 6, 2008
Published electronically: August 3, 2009
Additional Notes: The first author’s work was supported in part by the Programme for New Century Excellent Talents in University (NCET-06-0712), the National Natural Science Foundation of China (NSFC 10871066, 10571053), and the Excellent Youth Project of Scientific Research Fund of Hunan Provincial Education Department (0513039)
The second author’s work was supported in part by the US National Science Foundation grant DMS-0612908.
Article copyright: © Copyright 2009 American Mathematical Society