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Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition

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Abstract

In this paper, some local and parallel finite element methods based on two-grid discretizations are provided and studied for the non-stationary Stokes-Darcy model with the Beavers-Joseph interface condition. Two local algorithms, the semi-discrete and fully discrete finite element algorithms, are first introduced and related error estimates are rigorously derived. Based upon the fully discrete local algorithm, two fully discrete parallel algorithms are subsequently developed. The backward Euler scheme is considered for the temporal discretization and finite element method is used for the spatial discretization. The main idea of the parallel algorithms is to solve a decoupled Stokes-Darcy model via a coarse grid on the whole domain, then solve residual equations with a finer grid on overlapped subdomains by some local and parallel procedures at each time step. Some local a priori error is also provided that is crucial to our theoretical analysis. Finally, some numerical results are reported to illustrate the validity of the algorithms.

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Acknowledgements

The authors would like to thank the reviewers for their constructive comments, which allowed for the improvement of the presentation of the results.

Funding

This work is subsidized by the National Natural Science Foundation of China (No. 12172202), the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (No. 2022KJ249), the Natural Science Foundation of Shandong Province (No. ZR2021MA063).

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

    Guangzhi Du, Shilin Mi & Xinhui Wang

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  1. Guangzhi Du
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Contributions

Guangzhi Du: conceptualization, formal analysis, writing, review. Shilin Mi: methodology, writing, review. Xinhui Wang: visualization, validation, review. All authors reviewed the manuscript.

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Correspondence to Guangzhi Du.

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Du, G., Mi, S. & Wang, X. Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition. Numer Algor 94, 1883–1918 (2023). https://doi.org/10.1007/s11075-023-01558-1

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