Year: 2007
Journal of Computational Mathematics, Vol. 25 (2007), Iss. 2 : pp. 185–200
Abstract
The convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order $L_2$ error bounds, and $2p+1$-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform $2p+1$-order superconvergence is observed numerically.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2007-JCM-8684
Journal of Computational Mathematics, Vol. 25 (2007), Iss. 2 : pp. 185–200
Published online: 2007-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Discontinuous Galerkin methods Singular perturbation Superconvergence Shishkin mesh Numerical traces.