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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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References

  1. Badea, L., Discacciati, M., Quarteroni, A.: Numerical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 115(2), 195–227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boubendir, Y., Tlupova, S.: Domain decomposition methods for solving Stokes-Darcy problems with bondary integrals. SIAM J. Sci. Comput. 434(1), B82–B106 (2013)

    Article  MATH  Google Scholar 

  3. Cai, M., Mu, M.: A multilevel decoupled method for a mixed Stokes/Darcy model. J. Comput. Appl. Math. 236(9), 2452–2465 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, M., Mu, M., Xu, J.: Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach. SIAM J. Numer. Anal. 47(5), 3325–3338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, M., Huang, P., Mu, M.: Some multilevel decoupled algorithms for a mixed Navier-Stokes/Darcy model. Adv. Comput. Math. 44, 115–145 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, W., Zhao, W.: Finite element approximation for Stokes-Darcy flow with the Beavers-Joseph interface conditions. SIAM J. Numer. Anal. 47(6), 4239–4256 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, W., Gunzburger, M., Hua, F., Wang, X.: A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system. SIAM J. Numer. Anal. 49(3), 1064–1084 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chidyagwai, P., Riviere, B.: On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Method. Appl. Mech. 198(47-48), 3806–3820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. Ph.D. dissertation École Polytechnique fédérale de Lausanne (2004)

  10. Du, G., Zuo, L.: Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta Math. Sci. 37(5), 1331–1347 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, G., Zuo, L.: Local and parallel finite element post-processing scheme for the Stokes problem. Comput. Math. Appl. 2017(1), 129–140 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Du, G., Zuo, L.: A parallel partition of unity scheme based on Two-Grid discretizations for the Navier-Stokes problem. J. Sci. Comput. 75(3), 1445–1462 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du, G., Zuo, L.: A two-grid parallel partition of unity finite element scheme. Numer. Algorithms 80(2), 429–445 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Du, G., Zuo, L.: A two-grid method with backtracking for the mixed Stokes/Darcy model. J. Numer. Math. 29(1), 39–46 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Du, G., Zuo, L.: Local and parallel finite element methods for the coupled Stokes/Darcy model. Numer. Algorithms 87(4), 1593–1611 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Du, G., Hou, Y., Zuo, L.: A modified local and parallel fintie element method for the mixed Stokes-Darcy model. J. Math. Anal Appl. 435 (2), 1129–1145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Du, G., Li, Q., Zhang, Y.: A two-grid method with backtracking for the mixed Navier-Stokes/Darcy model. Numer. Meth. Part. D. E. 36(6), 1601–1610 (2020)

    Article  MathSciNet  Google Scholar 

  18. Du, G., Zuo, L., Zhang, Y.: A new local and parallel finite element method for the coupled Stokes-Darcy model. J. Sci Comput. 90(1), 43 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. He, Y., Xu, J., Zhou, A.: Local and parallel finite element algorithms for the Navier-Stokes problem. J. Comput. Math. 24(3), 227–238 (2006)

    MathSciNet  MATH  Google Scholar 

  20. He, Y., Xu, J., Zhou, A., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109(3), 415–434 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. He, X., Li, J., Lin, Y., Ming, J.: A domain decomposition method for the steady-state Navier-Stokes-Darcy model with the Beavers-Joseph interface condition. SIAM J. Sci. Comput. 37(5), S264–S290 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Hou, Y.: Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes-Darcy model. Appl. Math. Lett. 57, 90–96 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hou, Y., Qin, Y.: On the solution of coupled Stokes/Darcy model with Beavers-Joseph interface condition. Comput. Math. Appl. 77(1), 50–65 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jiang, B.: A parallel domain decomposition method for coupling of surface and groundwarter flows. Compu. Method. Appl. M. 198(9–12), 947–957 (2009)

    Article  MATH  Google Scholar 

  26. Layton, W., Tobiska, L.: A two-level method withbacktracking for the Navier-Stokes equations. SIAM J. Numer. Anal. 35(5), 2035–2054 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Layton, W., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, Q., Du, G.: Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equations. Numer. Algorithms 88(4), 1915–1936 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, Y., Hou, Y.: Error estimates of a Second-Order decoupled scheme for the evolutionary Stokes-Darcy system. Appl. Numer. Math. 154(129–148), 129–148 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, R., Gao, Y., Yan, W., Chen, Z.: A Crank-Nicolson discontinuous finite volume element method for a coupled non-stationary Stokes-Darcy problem. J. Comput. Appl. Math. 353, 86–112 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mu, M., Xu, J.: A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45 (5), 1801–1813 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shan, L., Zheng, H.: Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with the Beavers-Joseph interface conditions. SIAM J. Numer. Anal. 51(2), 813–839 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shang, Y., He, Y.: Parallel iterative finite element algorithms based on full domain partition for the stationary Navier-Stokes equations. Appl. Numer. Math. 60(7), 719–737 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shang, Y., He, Y., Kim, D., Zhou, X.: A new parallel finite element algorithm for the stationary Navier-Stokes equations. Finite. Elem. Anal. Des. 47(11), 1262–1279 (2011)

    Article  MathSciNet  Google Scholar 

  35. Song, L., Li, P., Gu, Y., Fan, C.: Generalized finite difference method for solving stationary 2D and 3D stokes equations with a mixed boundary condition. Comput. Math. Appl. 80(6), 1726–1743 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, Y., Sun, W., Zheng, H.: Domain decomposition method for the fully-mixed Stokes-Darcy coupled problem. Compu. Method. Appl. Mech. 374, 113578 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, X., Du, G., Zuo, L.: A novel local and parallel finite element method for the mixed Navier-Stokes-Darcy problem. Comput. Math Appl. 90 (15), 73–79 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yu, J., Shi, F., Zheng, H.: Local and parallel finite element algorithms based on the partition of unity for the stokes problem. SIAM J. Sci. Comput. 36(5), C547–C567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zheng, B., Shang, Y.: Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier-Stokes equations. Appl. Math. Comput. 357, 35–56 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Zheng, H., Yu, J., Shi, F.: Local and parallel finite element method based on the partition of unity for incompressible flow. J. Sci. Comput. 65 (2), 512–532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zheng, H., Shi, F., Hou, Y., Zhao, J., Cao, Y., Zhao, R.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal Appl. 435(1), 1–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zuo, L., Du, G.: A multi-grid technique for coupling fluid flow with porous media flow. Comput. Math. Appl. 75(11), 4012–4021 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zuo, L., Du, G.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer Algorithms 77(1), 151–165 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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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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