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Local and parallel partition of unity scheme for the mixed Navier-Stokes-Darcy problem

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Abstract

In this paper, by combining the two-grid method with a partition of unity, a local and parallel partition of unity scheme is designed and investigated for the mixed Navier-Stokes-Darcy problem. The main features of the present method are the following: (1) Once a coarse solution is derived, both the linearized Navier-Stokes and Darcy subproblems with a finer grid are independent, which allows a parallel computing with little communication; (2) a partition of unity is considered to assemble the solutions obtained from the local subdomains to arrive at a global continuous approximation; (3) a further coarse grid correction is carried out to derive optimal error bounds for the fluid velocity and piezometric head in L2-norm. Moreover, the convergence of the proposed method is shown. Some numerical experiments are reported to demonstrate the theoretical results.

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Funding

This work is subsidized by the National Natural Science Foundation of China (Nos. 12172202, 12001234, 11701343), the Natural Science Foundation of Shandong Province (No. ZR2021MA063), the Natural Science Foundation of Shaanxi Province (2021JQ-426) and the Scientific Research Program of Shaanxi Provincial Education Department (21JK0935).

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Authors and Affiliations

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

    Guangzhi Du

  2. School of Mathematical Sciences, University of Jinan, Jinan, 250022, China

    Liyun Zuo

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  1. Guangzhi Du
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  2. Liyun Zuo
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Correspondence to Liyun Zuo.

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Du, G., Zuo, L. Local and parallel partition of unity scheme for the mixed Navier-Stokes-Darcy problem. Numer Algor 91, 635–650 (2022). https://doi.org/10.1007/s11075-022-01276-0

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