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Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer


Authors: Huiqing Zhu and Zhimin Zhang
Journal: Math. Comp. 83 (2014), 635-663
MSC (2010): Primary 65N30, 65N12, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2013-02736-6
Published electronically: June 25, 2013
MathSciNet review: 3143687
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Abstract: In this paper, we study a uniform convergence property of the local discontinuous Galerkin method (LDG) for a convection-diffusion problem whose solution has exponential boundary layers. A Shishkin mesh is employed. The trail functions in the LDG method are piecewise polynomials that lies in the space $ \mathcal {Q}_k$, i.e., are tensor product polynomials of degree at most $ k$ in one variable, where $ k>0$. We identify that the error of the LDG solution in a DG-norm converges at a rate of $ (\ln N/N)^{k+1/2}$; here the total number of mesh points is $ O(N^2)$. The numerical experiments show that this rate of convergence is sharp.


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Additional Information

Huiqing Zhu
Affiliation: Department of Mathematics, The University of Southern Mississippi, Hattiesburg, Mississippi 39406
Email: Huiqing.Zhu@usm.edu

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202; Beijing Computational Science Research Center, No. 3 Heqing Road, Haidian District, Beijing 100084, China.
Email: zzhang@math.wayne.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02736-6
Keywords: Singularly perturbed, discontinuous Galerkin method, Shishkin mesh, uniform convergence.
Received by editor(s): April 2, 2011
Received by editor(s) in revised form: June 15, 2012
Published electronically: June 25, 2013
Additional Notes: Corresponding author: Huiqing Zhu, Huiqing.Zhu@usm.edu
The second author was supported in part by the US National Science Foundation through grant DMS-1115530.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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