我們對斯托克斯與達西的耦合系統提出一個有效率的演算法，這邊的耦合系統是指自由的流體(斯托克斯區域)與多孔材質的滲流(達西區域)之間的流體行為。文章內使用著名的「比弗斯-約瑟夫-薩夫曼」條件作為系統的交面條件。我們使用這個條件可以讓耦合系統在新的弱形式離散化時不需再加上額外的條件。這個演算法是建立在一個交面上非顯然的預處理算子上。這個預處理系統是均勻良態的，它跟網格的大小、黏度、滲透率無關。

We propose an efficient solver for the coupled Darcy-Stokes system modeling the composition of porous media region (Darcy) and free flow region (Stokes). An additional set of interface condition, known as the Beavers-Joseph-Saffman condition, is supplemented on the Darcy-Stokes interface. The coupled system is discretized based on a new weak formulation which incorporates the BJS condition naturally without additional regularity requirement. The proposed solver is based on a new preconditioning operator which has a nontrivial component on the interface. The preconditioned system is observed to be uniformly well conditioned independent of mesh size, viscosity or permeability.

References
[1] L. Badea, M. Discacciati, and A. Quarteroni. Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical Report 2, Technical Report of Politecnico di Milano, 2010.
[2] M. L. Bars and M. G. Worster. Interfacial conditions between a pure fluid and a porous medium-implications for binary alloy solidification. J. Fluid Mech., 550(1):149-173, 2006.
[3] C. Bernardi, F. Hecht, and F. Z. Nouri. A new finite-element discretization of the Stokes problem coupled with the Darcy equations. IMA J. Numer. Anal., 30(1):61-93, 2008.
[4] Y. Cao, M. Gunzburger, X. Hu, F. Hua, X. Wang, and W. Zhao. Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions. SIAM Journal on Numerical Analysis, 47(6):4239-4256, 2010.
[5] Y. Cao, M. Gunzburger, F. Hua, and X.Wang. Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Comm. Math. Sci, 8(1):1-25, 2008.
[6] Q. V. Dinh, R. Glowinski, J. Periaux, and G. Terrasson. On the coupling of viscous and inviscid models for incompressible fluid flows via domain decomposition. pages 350-369. Society for Industrial and Applied, 1988.
[7] M. Discacciati, E. Miglio, and A. Quarteroni. Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math., 43(1-2):57-74, 2002.
[8] M. Discacciati and A. Quarteroni. Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. Numerical Mathematics and Advanced Applications ENUMATH 2001, pages 3-20, 2003.
[9] M. Discacciati and A. Quarteroni. Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Vis. Sci., 6(2):93-103, 2004.
[10] M. Discacciati, A. Quarteroni, and A. Valli. Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Num. Anal., 45(3):1246-1268, 2007.
[11] F. El Chami, G. Mansour, and T. Sayah. Error studies of the coupling Darcy-Stokes system with velocity-pressure formulation. Calcolo, 49(2):73-93, 2011.
[12] F. El Dabaghi and O. Pironneau. Stream vectors in three dimensional aerodynamics. Numerische Mathematik, 48(5):561-589, 1986.
[13] V. J. Ervin, E. W. Jenkins, and S. Sun. Coupled generalized non-linear Stokes flow with flow through a porous media. SIAM J. Numer. Anal, 47(2):929-952, 2009.
[14] J. Galvis and M. Sarkis. Balancing domain decomposition methods for mortar coupling Stokes-Darcy systems. Domain decomposition methods in science and engineering XVI, 55:373-380, 2007.
[15] J. Galvis and M. Sarkis. Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal., 26:350-384, 2007.
[16] V. Girault and P. A. Raviart. Finite element methods for Navier-Stokes equations: theory and algorithms. Springer Science and Business Media, 1986.
[17] V. Girault and B. Riviere. DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal., 47(3):2052-2089, 2009.
[18] P. Grisvard. Elliptic problems in nonsmooth domains. Pitman Advanced Publishing Program, 1985.
[19] N. S. Hanspal, A. N. Waghode, V. Nassehi, and R. J. Wakeman. Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media, 64(1):73-101, 2006.
[20] X. He, J. Li, Y. Lin, and J. Ming. A domain decomposition method for the steady-state Navier-Stokes-Darcy model with Beavers-Joseph interface condition. SIAM Journal on Scientific Computing, 37(5):S264-S290, 2015.
[21] M. Hellou, J. Martinez, and M. El Yazidi. Stokes flow through microstructural model of fibrous media. Mech. Res. Commun., 31(1):97-103, 2004.
[22] Z. J. Huang and J. M. Tarbell. Numerical simulation of mass transfer in porous media of blood vessel walls. Am. J. Physiol. Heart Circ. Physiol., 273(1):H464-H477, 1997.
[23] O. Iliev and V. Laptev. On numerical simulation of flow through oil filters. Comput. Vis. Sci., 6(2):139-146, 2004.
[24] W. Jager and A. Mikelic. On the boundary condition at the contact interface between a porous medium and a free fluid. Ann. Scuola Norm-Sci., 23:403-465, 1996.
[25] W. Jager and A. Mikelic. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math., 60(4):1111-1127, 2000.
[26] A. R. Khaled and K. Vafai. The role of porous media in modeling flow and heat transfer in biological tissues. Int. J. Heat and Mass Transfer, 46(26):4989-5003, 2003.
[27] J. Koelman and M. Nepveu. Darcy flow in porous media: Cellular automata simulations. Numerical Methods for the Simulation of Multi-Phase and Complex Flow, 398:136-145, 1992.
[28] K. Lipnikov, D. Vassilev, and I. Yotov. Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes-Darcy flows on polygonal and polyhedral grids. Numer. Math., pages 321-360, 2014.
[29] Y. Maday, D. Meiron, A. T. Patera, and E. M. Ronquist. Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM J. Sci. Comput., 14(2):310-337, 1993.
[30] K. A. Mardal, X. C. Tai, and R. Winther. A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal., 40(5):1605-1631, 2003.
[31] M. Mu and J. Xu. A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM journal on numerical analysis, 45(5):1801-1813, 2007.
[32] D. A. Nield and A. Bejan. Convection in Porous Media. Springer Verlag, 1999.
[33] B. Riviere. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput., 22-23:479-500, 2005.
[34] B. Riviere and I. Yotov. Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal., 42:1959-1977, 2005.
[35] J. M. Urquiza, D. N'Dri, A. Garon, and M. C. Delfour. Coupling Stokes and Darcy equations. Appl. Numer. Math., 58(5):525-538, 2007.
[36] W. Wang and C. Xu. Spectral methods based on new formulations for coupled Stokes and Darcy equations. J. Comput. Phys., 257:126-142, 2014.
[37] W.-C. Wang and W. Wang. work in preparation.
[38] X. Xie, J. Xu, and G. Xue. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. J. Comput. Math., 26(3):437-455, 2008.
[39] C. Xu and Y. Lin. Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation. Int. J. Numer. Meth. Fluids, 32(6):619-646, 2000.