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Volume 25, Issue 2
Superconvergence of DG Method for One-Dimensional Singularly Perturbed Problems

Ziqing Xie & Zhimin Zhang

J. Comp. Math., 25 (2007), pp. 185-200.

Published online: 2007-04

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  • 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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@Article{JCM-25-185, author = {}, title = {Superconvergence of DG Method for One-Dimensional Singularly Perturbed Problems}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8684.html} }
TY - JOUR T1 - Superconvergence of DG Method for One-Dimensional Singularly Perturbed Problems JO - Journal of Computational Mathematics VL - 2 SP - 185 EP - 200 PY - 2007 DA - 2007/04 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8684.html KW - Discontinuous Galerkin methods, Singular perturbation, Superconvergence, Shishkin mesh, Numerical traces. AB -

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.

Ziqing Xie & Zhimin Zhang. (1970). Superconvergence of DG Method for One-Dimensional Singularly Perturbed Problems. Journal of Computational Mathematics. 25 (2). 185-200. doi:
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