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A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

  • Peiqi Huang (a1) and Zhilin Li (a2)

A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

Corresponding author
*Corresponding author. Email addresses: (P. Q. Huang), (Z. Li)
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[1] T. Arbogast and M. Gomez , A discretization and multigrid solver for a Darcy-Stokes system of three dimensional vuggy porous media, Comput. Geosci., 13 (2009), 331348.

[2] I. Babuska and G. N. Gatica , A residual-based a posteriori error estimator for the Stokes-Darcy coupled problem, SIAM J. Numer. Anal., 48 (2010), 498523.

[3] G. S. Beavers and D. D. Joseph , Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197207.

[4] F. Ben Belgacem , The mixed mortar finite element method for the incompressible Stokes problem: convergence analysis, SIAM J. Numer. Anal., 37 (2000), 10851100.

[5] C. Bernardi , T. C. Rebollo , F. Hecht and Z. Mghazli , Mortar finite element discretization of a model coupling Darcy and Stokes equations, Math. Model. Numer. Anal., 42 (2008), 375410.

[6] J. M. Boland and R. A. Nicolaides , Stability of finite elements under divergence constraints, SIAM J. Numer. Anal., 20 (1983), 722731.

[7] S. C. Brenner , Poincaré-Friedrichs inequalities for piecewise H 1 functions, SIAM J. Numer. Anal., 41 (2003), 306324.

[8] F. Brezzi and M. Fortin , Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.

[9] E. Burman and P. Hansbo , A unified stabilized method for Stokes’ and Darcy's equations, J. Comput. Appl. Math., 198 (2007), 3551.

[10] M. C. Cai , M. Mu and J. C. Xu , Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., 233 (2009), 346355.

[11] W. B. Chen , M. Gunzburger , F. Hua and X. M. Wang , A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system, SIAM J. Numer. Anal., 49 (2011), 10641084.

[12] C. D’Angelo and P. Zunino , A finite element method based on weighted interior penalties for heterogeneous incompressible flows, SIAM J. Numer. Anal., 47 (2009) 39904020.

[13] M. Discacciati , E. Miglio and A. Quarteroni , Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2002), 5774.

[14] A. Ern and J.-L. Guermond , Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004.

[15] M. F. Feng , R. S. Qi , R. Zhu and B. T. Ju , Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem, Appl. Math. Mech. -Engl. Ed., 31 (2010), 393404.

[17] G. N. Gatica , R. Oyarzúa and F. J. Sayas , Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp., 80 (2011), 19111948.

[18] V. Girault and P. A. Raviart , Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin, New York, 1986.

[19] P. Q. Huang and J. R. Chen , Two-level and multilevel methods for Stokes-Darcy problem discretized by nonconforming elements on nonmatching meshes (in Chinese), Sci. Sin. Math., 42 (2012), 389402.

[20] P. Q. Huang , J. R. Chen and M. C. Cai , A mixed and nonconforming FEM with nonmatching meshes for a coupled Stokes-Darcy model, J. Sci. Comput., 53 (2012), 377394.

[21] W. J. Layton , F. Schieweck and I. Yotov , Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2003), 21952218.

[23] K. A. Mardal , X. C. Tai and R. Winther , A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal., 40 (2002), 16051631.

[24] M. Mu and J. C. 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), 18011813.

[25] B. Riviere and I. Yotov , Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal., 42 (2005), 19591977.

[26] H. Rui and R. Zhang , A unified stabilized mixed finite element method for coupling Stokes and Darcy flows, Comput. Methods Appl. Mech. Engrg., 198 (2009), 26922699.

[27] P. Saffman , On the boundary condition at the surface of a porous media, Stud. Appl. Math., 50 (1971), 93101.

[31] X. J. Xu and S. Y. Zhang , A new divergence-free interpolation operator with applications to the Darcy-Stokes-Brinkman equations, SIAM J. Sci. Comput., 32 (2010), 855874.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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