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

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- Math. Comp.
**83**(2014), 635-663 Request permission

## 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.## References

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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
- 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. - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 635-663 - MSC (2010): Primary 65N30, 65N12, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-2013-02736-6
- MathSciNet review: 3143687