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Local and Parallel Finite Element Methods Based on Two-grid Discretizations for a Transient Coupled Navier-Stokes/Darcy Model

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Abstract

In this paper, some local and parallel finite element methods based on two-grid methods are presented for the non-stationary Navier-Stokes/Darcy model. Based on two-grid methods for spatial discretizations, both semi-discrete scheme and fully-discrete scheme with backward Euler method for the temporal discretization are proposed. Some local a priori estimate, which is crucial for the theoretical analysis, is obtained. The motivation of these local and parallel methods is that by utilizing decoupled method based on interface approximation via temporal extrapolation, low frequency could be obtained on the whole domain with a coarse grid, then solve some residual equations on some overlapped subdomains with a finer gird by some local and parallel procedures at each time step to catch high frequency. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations on the previous time step. To overcome the global discontinuity of the numerical solution generated by the local and parallel finite element algorithms, a new parallel algorithm based on the partition of unity is developed. In the end, some numerical experiments are constructed to prove the effectiveness of our algorithms.

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References

  1. 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, 1801–1813 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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, 3325–3338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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, 1601–1610 (2020)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Qin, Y., Hou, Y.: Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Navier-Stokes/Darcy Model. Acta. Math. Sci. 38B, 1361–1369 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Discacciati, M., Quarteroni, A., Valli, A.: Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45, 1246–1268 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

  11. 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, 1064–1084 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cao, Y., Gunzburger, M., He, X., Wang, X.: Robin-Robin domain decomposition methods for the steady-state Stokes-Darcy system with Beaver-Joseph interface condition. Numer. Math. 117, 601–629 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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, S264–S290 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Vassilev, D., Wang, C., Yotov, I.: Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Engrg. 268, 264–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sun, Y., Sun, W., Zheng, H.: Domain decomposition method for the fully-mixed Stokes-Darcy coupled problem. Comput. Methods Appl. Engrg. 374, 113578 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Gatica, G., Oyarzúa, R., Sayas, F.J.: A conforming mixed finite element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 29, 86–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron T. Numer. Ana. 26, 350–384 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. 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, 1331–1347 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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, 73–79 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Mu, M., Zhu, X.: Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comp. 79, 707–731 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, J., Li, R., Zhao, X., Chen, Z.: A second-order fractional time-stepping method for a coupled Stokes/Darcy system. J. Comput. Appl. Math. 390, 113329 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qin, Y., Hou, Y., Huang, P., Wang, Y.: Numerical analysis of two grad-div stabilization methods for the time-dependent Stokes/Darcy model. Comput. Math. Appl. 79, 817–832 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shan, L., Zhang, Y.: Error estimates of the partitioned time stepping method for the evolutionary Stokes-Darcy flows. Comput. Math. Appl. 73, 713–726 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shan, L., Zheng, H., Layton, W.: A decoupling method with different subdomain time steps for the nonstationary stokes-darcy model. Numer. Meth. Part. D. E. 29, 549–583 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, Y., Hou, Y., Layton, W., Zhao, Ha.: Adaptive Partitioned method for the Time-Accurate Approximation of the Evolutionary Stokes-Darcy System. Comput. Method. Appl. M. 364, 112923 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xue, D., Hou, Y.: Numerical Analysis of a Second Order Algorithm for a Non-stationary Navier-Stokes/Darcy Model. J. Comput. Appl. Math. 369, 112579 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Cao, L., He, Y., Li, J., Yang, D.: Decoupled modified characteristic FEMs for fully evolutionary Navier-Stokes-Darcy model with the Beavers-Joseph interface condition. J. Comput. Appl. Math. 38, 113128 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Du, G.: Expandable parallel finite element methods for linear elliptic problems. Acta Math. Sci. 40B, 572–588 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Du, G., Zuo, L.: A Parallel Iterative Finite Element Method for the Linear Elliptic Equations. J. Sci. Comput. 85, 35 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hou, Y., Du, G.: An expandable local and parallel two-grid finite element scheme. Comput. Math. Appl. 71, 2541–2556 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14, 293–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Du, G., Zuo, L.: Local and parallel finite element post-processing scheme for the Stokes problem. Comput. Math. Appl. 73, 129–140 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. 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, C547–C567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Du, G., Zuo, L.: A Parallel Partition of Unity Scheme Based on Two-Grid Discretizations for the Navier-Stokes Problem. J. Sci. Comput. 75, 1445–1462 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ran, H., Zheng, B., Shang, Y.: A parallel finite element variational multiscale method for the Navier-Stokes equations with nonlinear slip boundary conditions. Appl. Numer. Math. 168, 274–292 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Zheng, B., Shang, Y.: A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations. Appl. Numer. Math. 159, 138–158 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  46. 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, 512–532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, F., Huang, Q.: Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems. J. Sci. Comput. 82, 20 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhang, Y., Hou, Y., Shan, L., Dong, X.: Local and Parallel Finite Element Algorithm for Stationary Incompressible Magnetohydrodynamics. Numer. Meth. Part. D. E. 33, 1513–1539 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Li, Q., Du, G.: Local and parallel finite element methods based on two-grid discretizations for unsteady convection-diffusion problem. Numer. Meth. Part. D. E. 37, 3023–3041 (2021)

    Article  MathSciNet  Google Scholar 

  50. Liu, Q., Hou, Y.: Local and parallel finite element algorithms for time-dependent convection-diffusion equations. Appl. Math. Mech. -Engl. Ed. 30, 787–794 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shang, Y., Wang, K.: Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer. Algor. 54, 195–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  53. Beavers, G., Joseph, D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

    Article  Google Scholar 

  54. Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  55. Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22, 2006–2028 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. Payne, L., Straughan, B.: Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modeling questions. J. Math. Pure Appl. 77, 317–354 (1998)

    Article  MATH  Google Scholar 

  58. Saffman, P.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 1, 93–101 (1971)

    Article  MATH  Google Scholar 

  59. Girault, V., Raviart, P.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1981)

    MATH  Google Scholar 

  60. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem III: smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25, 489–512 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  62. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem IV: error analysis for second-order time discretization. SIAM. J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  63. He, Y.: A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations II: time discretization. J. Comput. Math. 22, 33–54 (2004)

    MathSciNet  MATH  Google Scholar 

  64. He, Y.: Two-level method baesd on finite element and crank-nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM. J. Numer. Anal. 41, 1263–1285 (2006)

    Article  Google Scholar 

  65. He, Y.: The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77, 2097–2124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work is subsidized by NSFC (Grant No. 12172202, 11701343), the Natural Science Foundation of Shandong Province (Grant 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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Correspondence to Guangzhi Du.

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Li, Q., Du, G. Local and Parallel Finite Element Methods Based on Two-grid Discretizations for a Transient Coupled Navier-Stokes/Darcy Model. J Sci Comput 92, 76 (2022). https://doi.org/10.1007/s10915-022-01946-0

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