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A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

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Abstract

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

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References

  1. Adams R-A. Sobolev Spaces. New York: Academic Press, 1975

    MATH  Google Scholar 

  2. Arbogast T, Gomez M. A discretization and multigrid solver for a Darcy-Stokes system of three dimensional vuggy porous media. Comput Geosci, 2009, 13: 331–348

    Article  MathSciNet  MATH  Google Scholar 

  3. Arbogast T, Wheeier M, Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J Numer Anal, 1997, 34: 828–852

    Article  MathSciNet  MATH  Google Scholar 

  4. Beavers S, Joseph D. Boundary conditions at a naturally permeable wall. J Fluid Mech, 1967, 30: 197–207

    Article  Google Scholar 

  5. Boubendir Y, Tlupova S. Stokes-Darcy boundary integral solutions using preconditioners. J Comput Phys, 2009, 228: 8627–8641

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenner S-C. Korn’s inequalities for piecewise H1 vector fields. Math Comput, 2003, 73: 1067–1087

    Article  MATH  Google Scholar 

  7. Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev Française Automat Informat Recherche Opérationnelle Sér Rouge, 1974, 8: 129–151

    MathSciNet  MATH  Google Scholar 

  8. Brezzi F, Douglas J, Fortin J-M, et al. Efficient rectangular mixed finite elements in two and three space variables. M2AN Math Model Numer Anal, 1987, 21: 581–604

    Article  MathSciNet  MATH  Google Scholar 

  9. Brezzi F, Douglas J, Marini L-D. Two families of mixed finite elements for second order elliptic problems. Numer Math, 1985, 88: 217–235

    Article  MathSciNet  MATH  Google Scholar 

  10. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag, 1991

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen W, Wang F, Wang Y. Weak Galerkin method for the coupled Darcy-Stokes flow. IMA J Numer Anal, 2016, 36: 897–921

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gartling D-K, Hickox C-E, Givler R-C. Simulation of coupled viscous and porous flow problems. Int J Comput Fluid Dyn, 1996, 7: 23–48

    Article  MATH  Google Scholar 

  16. Girault V, Kanschat G, Rivière B. Error analysis for a monolithic discretization of coupled Darcy and Stokes problems. J Numer Math, 2014, 22: 109–142

    Article  MathSciNet  MATH  Google Scholar 

  17. Hanspal N, Waghode A, Nassehi V, et al. Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp Porous Media, 2006, 64: 73–101

    Article  Google Scholar 

  18. Hong Q, Wang F, Wu S, et al. A unified study of continuous and discontinuous Galerkin methods. Sci China Math, 2019, 62: 1–32

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong Q, Xu J. Uniform stability and error analysis for some discontinuous Galerkin methods. J Comput Math, 2021, 39: 283–310

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Kanschat G, Rivière B. A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J Comput Phys, 2010, 229: 5933–5943

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Layton W, Tran H, Trenchea C. Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows. SIAM J Numer Anal, 2013, 51: 248–272

    Article  MathSciNet  MATH  Google Scholar 

  24. Li R, Gao Y, Li J, et al. A weak Galerkin finite element method for a coupled Stokes-Darcy problem on general meshes. J Comput Appl Math, 2018, 334: 111–127

    Article  MathSciNet  MATH  Google Scholar 

  25. Li R, Li J, Liu X, et al. A weak Galerkin finite element method for a coupled Stokes-Darcy problem. Numer Methods Partial Differential Equations, 2017, 33: 1352–1373

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin G, Liu J, Sadre-Marandi F. A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. J Comput Appl Math, 2015, 273: 346–362

    Article  MathSciNet  MATH  Google Scholar 

  27. Lions J-L, Magenes E. Non-Homogeneous Boundary Value Problems and Applications. New York: Springer-Verlag, 1972

    Book  MATH  Google Scholar 

  28. Masud A, Hughes T. A stabilized mixed finite element method for Darcy flow. Comput Methods Appl Mech Engrg, 2002, 191: 4341–4370

    Article  MathSciNet  MATH  Google Scholar 

  29. Mu L, Wang J, Wang Y, et al. A computational study of the weak Galerkin method for second-order elliptic equations. Numer Algorithms, 2012, 63: 753–777

    Article  MathSciNet  MATH  Google Scholar 

  30. Mu L, Wang J, Ye X, et al. A weak Galerkin finite element method for the Maxwell equations. J Sci Comput, 2015, 65: 363–386

    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, 2009, 45: 1801–1813

    Article  MathSciNet  MATH  Google Scholar 

  32. Nassehi V. Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration. Chem Eng Sci, 1998, 53: 1253–1265

    Article  Google Scholar 

  33. Raviart R-A, Thomas J-M. A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606. New York: Springer-Verlag, 1977, 292–315

    Chapter  Google Scholar 

  34. Rivière B. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problem. J Sci Comput, 2005, 22: 479–500

    Article  MathSciNet  MATH  Google Scholar 

  35. Rivière B, Yotov I. Locally conservative coupling of Stokes and Darcy flow. SIAM J Numer Anal, 2005, 42: 1959–1977

    Article  MathSciNet  MATH  Google Scholar 

  36. Saffman P. On the boundary condition at the surface of a porous media. Stud Appl Math, 1971, 50: 292–315

    Article  Google Scholar 

  37. Salinger A-G, Aris R, Derby J-J. Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains. Internat J Numer Methods Fluids, 1994, 18: 1185–1209

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang C, Wang J, Wang R, et al. A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J Comput Appl Math, 2016, 307: 346–366

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math, 2013, 241: 103–115

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math Comp, 2014, 83: 2101–2126

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang J, Ye X. A weak Galerkin finite element method for the Stokes equations. Adv Comput Math, 2016, 42: 155–174

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang X, Zhai Q, Zhang R. The weak Galerkin method for solving the incompressible Brinkman flow. J Comput Appl Math, 2016, 307: 13–24

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhai Q, Zhang R, Mu L. A new weak Galerkin finite element scheme for the Brinkman model. Commun Coumput Phys, 2016, 19: 1409–1434

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang R, Zhai Q. A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J Sci Comput, 2015, 64: 559–585

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang T, Tao L. A stable weak Galerkin finite element method for Stokes problem. J Comput Appl Math, 2018, 333: 235–246

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11901015, 11971198, 91630201, 11871245, 11771179 and 11826101), and the Program for Cheung Kong Scholars (Grant No. Q2016067), Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Hui Peng, Qilong Zhai & Ran Zhang

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Hui Peng
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  2. Qilong Zhai
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  3. Ran Zhang
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  4. Shangyou Zhang
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Correspondence to Qilong Zhai.

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Peng, H., Zhai, Q., Zhang, R. et al. A weak Galerkin-mixed finite element method for the Stokes-Darcy problem. Sci. China Math. 64, 2357–2380 (2021). https://doi.org/10.1007/s11425-019-1855-y

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  • DOI: https://doi.org/10.1007/s11425-019-1855-y

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