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Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

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Abstract

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1) in the H1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1 + τ2) in the H1 norm can be obtained in the Crank-Nicolson fully discrete scheme.

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Acknowledgements

We would like to thank anonymous referees for their insightful comments that improved this paper.

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Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, Hunan Province, China

    Jianyun Wang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yanping Chen

Authors
  1. Jianyun Wang
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  2. Yanping Chen
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Correspondence to Yanping Chen.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11671157)

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Wang, J., Chen, Y. Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrödinger equation. Appl. Math. Mech.-Engl. Ed. 39, 1353–1372 (2018). https://doi.org/10.1007/s10483-018-2369-9

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  • DOI: https://doi.org/10.1007/s10483-018-2369-9

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