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Supercloseness of Continuous Interior Penalty Methods on Shishkin Triangular Meshes and Hybrid Meshes for Singularly Perturbed Problems with Characteristic Layers

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Abstract

A singularly perturbed convection–diffusion problem posed on the unit square is solved using a continuous interior penalty (CIP) method. The mesh used is a Shishkin triangular mesh or a Shishkin hybrid mesh consisting of triangles and rectangles. For the CIP method, a variant of Oswald interpolation operator is introduced for a discrete inf-sup stability, which is proved in a new norm stronger than the the usual CIP norm. This stability and a new cancellation technique enable new supercloseness results for the CIP method: the computed solutions on the triangular mesh and the hybrid mesh are shown to be 3/2 order and 2 order (up to a logarithmic factor) convergent in the new norm to the interpolants of the true solution, respectively. These convergence orders are uniformly valid with respect to the diffusion parameter and imply that for the Shishkin mesh the hybrid mesh is superior to the triangular one. Numerical experiments illustrate these theoretical results.

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Acknowledgements

We thank the two anonymous referees for their valuable comments and suggestions that led us to improve this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Jin Zhang

  2. College of Science, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaowei Liu

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  1. Jin Zhang
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  2. Xiaowei Liu
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Correspondence to Jin Zhang.

Additional information

Jin Zhang is supported by National Science Foundation of China (11771257,11501335), Shandong Provincial Natural Science Foundation, China (ZR2017MA003) and A Project of Shandong Province Higher Educational Science and Technology Program (J17KA169).

Xiaowei Liu is supported by National Science Foundation of China (11601251) and Shandong Provincial Natural Science Foundation, China (ZR2016AM13).

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Zhang, J., Liu, X. Supercloseness of Continuous Interior Penalty Methods on Shishkin Triangular Meshes and Hybrid Meshes for Singularly Perturbed Problems with Characteristic Layers. J Sci Comput 76, 1633–1656 (2018). https://doi.org/10.1007/s10915-018-0677-y

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  • DOI: https://doi.org/10.1007/s10915-018-0677-y

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