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Multigrid Algorithm for Immersed Finite Element Discretizations of Elliptic Interface Problems

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Abstract

This paper is devoted to analyzing multigrid algorithm for solving elliptic interface problems discretized using the partially penalized immersed finite element (PPIFE). By taking the average values of nodal variables and integral variables, we construct intergrid transfer operators for the \(P_1\) partially penalized immersed finite element (\(P_1\)-PPIFE) and the Crouzeix–Raviart partially penalized immersed finite element (CR-PPIFE), which satisfy certain stable approximation properties. An extra interface correction procedure is added in the smoothing steps to ensure the robustness of multigrid algorithm. We prove that the convergence of W-cycle multigrid algorithm and the condition number of the variable V-cycle as preconditioner are optimal by verifying the regularity-approximation assumption, which means that the convergence rate of algorithm is independent of mesh level, mesh size, and the position of the interface relative to the mesh. Numerical experiments illustrate the convergence of our algorithms using the W-cycle, V-cycle, and preconditioned conjugate gradient algorithm (PCG) with the V-cycle.

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Acknowledgements

The authors would like to thank the three anonymous reviewers for their helpful comments and suggestions, which greatly helped improve the quality of the paper.

Funding

This research is supported by the National Natural Science Foundation of China (Grant Nos. 12071227, 12201547 and 12371370) and NSAF (Grant No. U2230402).

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Authors and Affiliations

  1. Key Laboratory of NSLSCS (Nanjing Normal University), Ministry of Education, Jiangsu International Joint Laboratory of Big Data Modeling, Computing, and Application, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Hanyu Chu & Yongzhong Song

  2. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Haifeng Ji

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Ying Cai

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  1. Hanyu Chu
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  2. Yongzhong Song
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Correspondence to Ying Cai.

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Chu, H., Song, Y., Ji, H. et al. Multigrid Algorithm for Immersed Finite Element Discretizations of Elliptic Interface Problems. J Sci Comput 98, 26 (2024). https://doi.org/10.1007/s10915-023-02416-x

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  • DOI: https://doi.org/10.1007/s10915-023-02416-x

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