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An Isoparametric Finite Element Method for Time-fractional Parabolic Equation on 2D Curved Domain

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Abstract

This paper provides an efficient numerical scheme for approximating the solution of the time-fractional parabolic equation on 2D curved domain. Here, the solution of the problem exhibits a weak singularity at the initial time \(t=0\). The method is based on applying the L1 formula to approximate the Caputo time-fractional derivative and using the isoparametric finite element method to approximate the spatial direction. A fully discrete scheme is then constructed using numerical quadrature. Since \(\Omega _{h}\) differs from \(\Omega \), the error estimates of the boundary terms on the region between \(\Omega _{h}\) and \(\Omega \) and the effect of numerical quadrature are both considered. Afterward, the stability analysis and optimal error estimates for the fully discrete scheme are proved in detail. Finally, some numerical experiments are presented to verify the theoretical results.

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Funding

This work is supported by the National Natural Science Foundation of China (No. 12071101, 12171122), Shenzhen Science and Technology Program (No. RCJC20210609103755110), Fundamental Research Project of Shenzhen (No. JCYJ20190806143201649) and Fundamental Research Funds for the Central Universities (No. 2022FRFK060026).

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Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Zhixin Liu & Minghui Song

  2. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Hui Liang

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  1. Zhixin Liu
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Correspondence to Minghui Song.

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Liu, Z., Song, M. & Liang, H. An Isoparametric Finite Element Method for Time-fractional Parabolic Equation on 2D Curved Domain. J Sci Comput 99, 88 (2024). https://doi.org/10.1007/s10915-024-02556-8

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  • DOI: https://doi.org/10.1007/s10915-024-02556-8

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