Skip to main content
SpringerLink
Log in

A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, based on a two-grid method and a recent local and parallel finite element method, a parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem is proposed and analyzed. This method ensures that all the local subproblems on the fine grid can be solved in parallel. Optimal error bounds of the approximate solution are obtained. Finally, numerical experiments are presented to demonstrate the accuracy and effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  3. Boubendir, Y., Tlupova, S.: Domain decomposition methods for solving Stokes-Darcy problems with bondary integrals. SIAM J. Sci. Comput. 35, B82–B106 (2013)

    Article  MATH  Google Scholar 

  4. Chen, W., Chen, P., Gunzburger, M., Yan, N.: Superconvergence analysis of FEMs for the Stokes-Darcy system. Math. Methods. Appl. Sci. 33, 1605–1617 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, W., Gunzburger, M., Hua, F., Wang, X.M.: 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 

  6. Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, X., Zhao, W.: Finite element approximation for Stokes-Darcy flow with the Beavers-Joseph interface conditions. Comm. Math. Sci. 8, 1–25 (2010)

    Article  MATH  Google Scholar 

  7. Cai, M.C., 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. Cai, M.C., Mu, M., Xu, J.C.: Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications. J. Comput. Appl. Math. 233, 346–355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cai, M.C., Mu, M., Xu, J.C.: 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 

  10. Cesmelioglu, A., Rivière, B.: Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J. Sci. Comput. 40, 115–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chidyagwai, P., Rivière, B.: On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198, 3806–3820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chidyagwai, P., Rivière, B.: A two-grid method for coupled free flow with porous media flow. Adv. Water Resour. 34, 1113–1123 (2011)

    Article  Google Scholar 

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

  14. Discacciati, M., Quarteroni, A.: Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Vis. Sci. 6, 93–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Du, G.Z., Hou, Y.R.: A parallel partition of unity method for the time-dependent convection-diffusion equations. Int. J. Numer. Method. H 25, 1947–1956 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Du, G.Z., Hou, Y.R., Zuo, L.Y.: Local and parallel finite element methods for the mixed Navier-Stokes/Darcy model. Int. J. Comput. Math., 1–18 (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Ervin, V.J., Jenkins, E.W., Sun, S.: Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47, 929–952 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feng, M., Qi, R., Zhu, R., Ju, B.: Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem. Appl. Math. Mech. 31, 393–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Girault, B.R.: Rivière DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Nuer. Anal. 47, 2052–2089 (2009)

    Article  MATH  Google Scholar 

  22. Galvis, J., Sarkis, M.: Balancing domain decomposition methods for mortar coupling Stokes-Darcy Systems, Proceedings of the 16th International Conference on Domain Decomposition Methods (2005)

  23. Glowinski, R., Pan, T., Periaux, J.: A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving grid bodies: I. Case where the rigid body motions are known a priori. C. R. Acad. Sci. Paris Ser. I Math. 324, 361–369 (1997)

    Article  MATH  Google Scholar 

  24. He, X.M., Li, J., Liu, Y.P., 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Jiang, B.: A parallel domain decomposition method for coupling of surface and groundwarter flows. Comput. Methods Appl. Mech. Eng. 198, 947–957 (2009)

    Article  MATH  Google Scholar 

  28. Kanschat, G., Rivière, B.: A strongly conservative finite element method for the coupling f Stokes and Darcy flow. J. Commput. Phys. 229, 5933–5943 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, Q.F., Hou, Y.R.: 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 

  30. Li, R., Li, J., Chen, Z.X., Gao, Y.L.: A stabilized finite element method based on two local Gauss integrations for a coupled Stokes-Darcy problem. J. Comput. Appl. Math. 292, 92–104 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Ma, F., Ma, Y., Wo, W.: Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations. Appl. Math. Mech. 28, 27–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Markus, S., Houstis, E., Catlin, A., Rice, J., Tsompanopoulou, P., Vavalis, E., Gottfried, D., Su, K., Balakrishnan, G.: An agent-based netcentric framework for multidisciplinary problem solving environments (MPSE). Int. J. Comput. Engrg. Sci. 1, 33–60 (2000)

    Article  Google Scholar 

  34. Mu, M.: Solving composite problems with interface relaxation. SIAM J. Sci. Comput. 20, 1394–1416 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Marquez, A., Meddahi, S., Sayas, F.J.: A decoupled preconditioning technique for a mixed Stokes-Darcy model. J. Sci. Comput. 57, 174–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mu, M., Xu, J.C.: 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 

  37. Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 93–101 (1971)

    Article  MATH  Google Scholar 

  38. Rivière, B.: Analysis of a discontinuous finite element methods for the coupled Stokes-and Darcy problems. J. Sci. Comput. 22, 1959–1977 (205)

  39. Rivière, B., Yotov, I.: Locally conservative coupling of Stokes-and Darcy flows. SIAM J. Numer. Anal. 42, 1959–1977 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, J.C., Zhou, A.H.: Some local and parallel properties of finite element discretizations. Proceedings for Eleventh International Conference on Domain Decomposition Methods, 140–147 (1999)

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Zuo, L.Y., Hou, Y.R.: A decoupling two-grid algorithm for the mixed Stokes-Darcy model with the Beavers-Joseph interface condition. Numer. Methods Partial Differ. Eqns. 3, 1066–1082 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zuo, L.Y., Hou, Y.R.: Numerical analysis for the mixed Navier-Stokes and Darcy Problem with the Beavers-Joseph interface condition. Numer. Methods Partial Differ. Eqns. 31, 1009–1030 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zuo, L.Y., Hou, Y.R.: A two-grid decoupling method for the mixed Stokes-Darcy model. J. Comput. Appl. Math. 275, 139–147 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

Subsidized by NSFC (Grant Nos. 11571274 and 11401466)

Author information

Authors and Affiliations

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

    Liyun Zuo

  2. School of Mathematical Sciences and Statistics, Shandong Normal University, Jinan, 250014, China

    Guangzhi Du

Authors
  1. Liyun Zuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangzhi Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liyun Zuo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zuo, L., Du, G. A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer Algor 77, 151–165 (2018). https://doi.org/10.1007/s11075-017-0308-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0308-y

Keywords

Search

Navigation