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Volume 27, Issue 2-3
A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems

Ziqing Xie, Zuozheng Zhang & Zhimin Zhang

J. Comp. Math., 27 (2009), pp. 280-298.

Published online: 2009-04

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  • Abstract

In this paper, we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform $2p + 1$-order superconvergence is observed numerically for both one-dimensional and two-dimensional cases.

  • Keywords

Singularly perturbed problems, Local discontinuous Galerkin method, Numerical fluxes, Uniform superconvergence.

  • AMS Subject Headings

65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-280, author = {Ziqing Xie, Zuozheng Zhang and Zhimin Zhang}, title = {A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {280--298}, abstract = {

In this paper, we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform $2p + 1$-order superconvergence is observed numerically for both one-dimensional and two-dimensional cases.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8573.html} }
TY - JOUR T1 - A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems AU - Ziqing Xie, Zuozheng Zhang & Zhimin Zhang JO - Journal of Computational Mathematics VL - 2-3 SP - 280 EP - 298 PY - 2009 DA - 2009/04 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8573.html KW - Singularly perturbed problems, Local discontinuous Galerkin method, Numerical fluxes, Uniform superconvergence. AB -

In this paper, we consider the local discontinuous Galerkin method (LDG) for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform $2p + 1$-order superconvergence is observed numerically for both one-dimensional and two-dimensional cases.

Ziqing Xie, Zuozheng Zhang and Zhimin Zhang. (2009). A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems. Journal of Computational Mathematics. 27 (2-3). 280-298. doi:
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