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A two-grid method with backtracking for the mixed Stokes/Darcy model

Guangzhi Du and Liyun Zuo

Abstract

In this paper, a two-grid method with backtracking is proposed and investigated for the mixed Stokes/Darcy system which describes a fluid flow coupled with a porous media flow. Based on the classical two-grid method [15], a coarse mesh correction is carried out to derive optimal error bounds for the velocity field and the piezometric head in L2 norm. Finally, results of numerical experiments are provided to support the theoretical results.

JEL Classification: 65N15; 65N30

Funding statement: This work is subsidized by NSFC(Grant No.11701343, 11801332) and Natural Science Foundation of Shandong Province (Grant No. ZR2019BA002, ZR2017BA027).

References

[1] Y. Boubendir and S. Tlupova, Domain decomposition methods for solving Stokes–Darcy problems with boundary integrals, SIAM J. Sci. Comput., 35 (2013), B82–B106. Search in Google Scholar

[2] M. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math., 236 (2012), 2452–2465. Search in Google Scholar

[3] W. Chen, M. Gunzburger, X. He, and X. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems, Math. Comp., 83 (2014), 1617–1644. Search in Google Scholar

[4] M. Discacciati and A. Quarteroni, Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations, Comput. Visual Sci., 6 (2004), 93–103. Search in Google Scholar

[5] M. Discacciati, A. Quarteroni, and A. Valli, Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246–1268. Search in Google Scholar

[6] G. Du, Y. Hou, and L. Zuo, A modified local and parallel finite element method for the mixed Stokes–Darcy model, J. Math. Anal. Appl., 435 (2016), 1129–1145. Search in Google Scholar

[7] G. Du and L. Zuo, Local and parallel finite element method for the mixed Navier–Stokes/Darcy model with Beavers–Joseph interface conditions, Acta Math. Sci., 37 (2017), 1331–1347. Search in Google Scholar

[8] P. Hessari and B. Shin, First order system least squares pseudo-spectral method for Stokes–Darcy equations, Appl. Numer. Math., 120 (2017), 35–52. Search in Google Scholar

[9] Y. Hou, Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes–Darcy model, Appl. Math. Letters, 57 (2016), 90–96. Search in Google Scholar

[10] Y. Hou and Y. Qin, On the solution of coupled Stokes/Darcy model with Beavers–Joseph interface condition, Comput. Math. Appl., 77 (2019), 50–65. Search in Google Scholar

[11] W. Layton, F. Schieweck, and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2003), 2195–2218. Search in Google Scholar

[12] Y. Li and Y. Hou, A second-order artificial compression method for the evolutionary Stokes–Darcy system, Numer. Algorithms, (2019), https://doi.org/10.1007/s11075-019-00791-x Search in Google Scholar

[13] A. Marquez, S. Meddahi, and F. Sayas, A decoupled preconditioning technique for a mixed Stokes–Darcy model, J. Sci. Comput., 57 (2013), 174–192. Search in Google Scholar

[14] M. Mu and X. Zhu, Decoupled schemes for a non-stationary mixed Stokes–Darcy model, Math. Comput., 79 (2010), 707–731. Search in Google Scholar

[15] M. Mu and J. Xu, A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801–1813. Search in Google Scholar

[16] Y. Qin and Y. Hou, Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Navier–Stokes/Darcy model, Acta Math. Sci., 38B (2018), 1361–1369. Search in Google Scholar

[17] L. Shan and H. Zheng, Partitioned time stepping method for fully evolutionary Stokes–Darcy flow with the Beavers–Joseph interface conditions, SIAM J. Numer. Anal., 51 (2013), 813–839. Search in Google Scholar

[18] L. Shan, H. Zheng, and W. Layton, A decoupling method with different subdomain time steps for the nonstationary Stokes–Darcy model, Numer. Meth. Part. D. E., 29 (2013), 549–583. Search in Google Scholar

[19] L. Zuo and G. Du, A multi-grid technique for coupling fluid flow with porous media flow, Comput. Math. Appl., 75 (2018), 4012–4021. Search in Google Scholar

[20] L. Zuo and G. Du, A parallel two-grid linearized method for the coupled Navier–Stokes–Darcy problem, Numer. Algorithms, 77 (2018), 151–165. Search in Google Scholar

[21] L. Zuo and G. Du, A multi-grid decoupling method for the coupled fluid flow with the porous media flow, J. Math. Fluid. Mech., 20 (2018), 683–695. Search in Google Scholar

[22] F. Hecht, New development in FreeFem++. J. Numer. Math., 20 (2012), 251–265. Search in Google Scholar

Received: 2019-02-06
Revised: 2020-04-19
Accepted: 2020-04-30
Published Online: 2020-05-04
Published in Print: 2021-03-26

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