Local and parallel finite element methods for the mixed Navier–Stokes/Darcy model

References

• R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
   Google Scholar
• T. Arbogast and D.S. Brunson, A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium, Comput. Geosci. 11 (2007), pp. 207–218. doi: 10.1007/s10596-007-9043-0
   Web of Science ®Google Scholar
• G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid. Mech. 30 (1967), pp. 197–207. doi: 10.1017/S0022112067001375
   Web of Science ®Google Scholar
• Y. Boubendir and S. Tlupova, Domain decomposition methods for solving Stokes–Darcy problems with bondary integrals, SIAM J. Sci. Comput. 35 (2013), pp. B82–B106. doi: 10.1137/110838376
   Google Scholar
• M.C. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math. 236 (2012), pp. 2452–2465. doi: 10.1016/j.cam.2011.12.003
   Web of Science ®Google Scholar
• M.C. Cai, M. Mu, and J.C. Xu, Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal. 47 (2009), pp. 3325–3338. doi: 10.1137/080721868
   Web of Science ®Google Scholar
• 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), pp. 346–355. doi: 10.1016/j.cam.2009.07.029
   Web of Science ®Google Scholar
• 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. J. Numer. Anal. 47 (2010), pp. 4239–4256. doi: 10.1137/080731542
   Web of Science ®Google Scholar
• Y. Cao, M. Gunzburger, F. Hua, and X. Wang, Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition, Commun. Math. Sci. 8 (2010), pp. 1–25. doi: 10.4310/CMS.2010.v8.n1.a2
   Web of Science ®Google Scholar
• Y.Z. Cao, M. Gunzburger, X.M. He, and X.M. Wang, Robin-Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition, Numer. Math. 117 (2011), pp. 601–629. doi: 10.1007/s00211-011-0361-8
   Web of Science ®Google Scholar
• Y.Z. Cao, M. Gunzburger, X.M. He, and X.M. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems, Math. Comput. 83 (2014), pp. 1617–1644. doi: 10.1090/S0025-5718-2014-02779-8
   Web of Science ®Google Scholar
• W. Chen and Y. Wang, A posteriori error estimate for the H(div) conforming mixed finite element for the coupled Darcy–Stokes system, J. Comput. Appl. Math. 255 (2014), pp. 502–516. doi: 10.1016/j.cam.2013.05.021
   Web of Science ®Google Scholar
• N. Chen, M. Gunzburger, and X. Wang, Asymptotic analysis of the differences between the Stokes–Darcy system with different interface conditions and the Stokes–Brinkman system, J. Math. Anal. Appl. 368 (2010), pp. 658–676. doi: 10.1016/j.jmaa.2010.02.022
   Web of Science ®Google Scholar
• W. 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), pp. 1064–1084. doi: 10.1137/080740556
   Web of Science ®Google Scholar
• P. Chidyagwai and B. Rivière, On the solution of the coupled Navier–Stokes and Darcy equations, Comput. Method Appl. M. 198 (2007), pp. 3806–3820. doi: 10.1016/j.cma.2009.08.012
   Web of Science ®Google Scholar
• P. Chidyagwai and B. Rivière, A two-grid method for coupled free flow with porous media flow, Adv. Water Resour. 34 (2011), pp. 1113–1123. doi: 10.1016/j.advwatres.2011.04.010
   Web of Science ®Google Scholar
• P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis, Vol. II. Finite Element Methods (Part I), North-Holland, Amsterdam, 1991.
   Google Scholar
• J.M. Connors and J.S. Howell, A fluid-fluid interaction method using decoupled subproblems and differing time steps, Numer. Methods Partial Differential Equations 28 (2012), pp. 1283–1308. doi: 10.1002/num.20681
   Web of Science ®Google Scholar
• M. Discacciati, Domain decomposition methods for the coupling of surface and groundwater flows, Ph.D. diss., École Polytechnique Fédérale de Lausanne, 2004.
   Google Scholar
• M. Discacciati and A. Quarteroni, Analysis of a domain decomposition method for the coupling Stokes and Darcy equations, in Numerical Analysis and Advanced Applications-Enumath 2001, F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, eds., Springer, Milan, 2003, pp. 3–20.
   Google Scholar
• 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 (2004), pp. 93–103. doi: 10.1007/s00791-003-0113-0
   Google Scholar
• M. Discacciati, E. Miglio, and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math. 43 (2002), pp. 57–74. doi: 10.1016/S0168-9274(02)00125-3
   Web of Science ®Google Scholar
• M. Discacciati, A. Quarteroni, and A. Valli, Robin-Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal. 45 (2007), pp. 1246–1268. doi: 10.1137/06065091X
   Web of Science ®Google Scholar
• W.Q. Feng, X.M. He, Z. Wang, and X. Zhang, Non-iterative domain decomposition methods for a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition, Appl. Math. Comput. 219 (2012), pp. 453–463. doi: 10.1016/j.amc.2012.05.012
   Web of Science ®Google Scholar
• J. Galvis and M. Sarkis, Balancing domain decomposition methods for mortar coupling Stokes–Darcy Systems, Proceedings of the 16th International Conference on Domain Decomposition Methods, New York, 2005.
   Google Scholar
• J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the coupling Stokes–Darcy equations, Electron. Trans. Numer. Anal. 26 (2007), pp. 350–384.
   Web of Science ®Google Scholar
• J. Galvis and M. Sarkis, FETI and BDD methods for mortar coupling of Stokes–Mortar–Darcy systems, Comm. Appl. Math. Comput. Sci. 5 (2010), pp. 1–30. doi: 10.2140/camcos.2010.5.1
   Web of Science ®Google Scholar
• V. Girault and B. Rivière, DG approximation of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition, SIAM J. Numer. Anal. 47 (2009), pp. 2052–2089. doi: 10.1137/070686081
   Web of Science ®Google Scholar
• V. Girault, D. Vassilev, and I. Yotov, Mortar multiscale finite element methods for Stokes–Darcy flows, Numer. Math. 127 (2014), pp. 93–165. doi: 10.1007/s00211-013-0583-z
   Web of Science ®Google Scholar
• R. Glowinski, T.-W. Pan, and J. Periaux, 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 (1997), pp. 361–369. doi: 10.1016/S0764-4442(99)80376-0
   Google Scholar
• Y.N He, Two-level method based on finite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations, SIAM J. Numer. Anal. 41 (2003), pp. 1263–1285. doi: 10.1137/S0036142901385659
   Web of Science ®Google Scholar
• Y.N. He, J.C. Xu, and A.H. Zhou, Local and parallel finite element algorithms for the Navier–Stokes problem, J. Comput. Math. 24 (2006), pp. 227–238.
   Web of Science ®Google Scholar
• Y.N. He, J.C. Xu, and A.H. Zhou, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), pp. 415–434. doi: 10.1007/s00211-008-0141-2
   Web of Science ®Google Scholar
• F. Hua, Modeling, analysis and simulation of Stokes–Darcy system with Beavers–Joseph interface condition, Ph.D. diss., The Florida State University, 2009.
   Google Scholar
• B. Jiang, A parallel domain decomposition method for coupling of surface and groundwater flows, Comput. Methods Appl. Mech. Engrg. 198 (2009), pp. 947–957. doi: 10.1016/j.cma.2008.11.001
   Web of Science ®Google Scholar
• W.J. Layton, F. Schieweck, and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 40 (2002), pp. 2195–2218. doi: 10.1137/S0036142901392766
   Web of Science ®Google Scholar
• Q.-F. Liu and Y.-R. Hou, Local and parallel finite element algorithms for time-dependent convection-diffusion equations, Appl. Math. Mech. (English Ed.) 30 (2009), pp. 787–794. doi: 10.1007/s10483-009-0613-x
   Web of Science ®Google Scholar
• F.-Y. Ma, Y.-C. Ma, and W.-F. Wo, Local and parallel finite element algorithms based on two-grid discretization for steady Navier–Stokes equations, Appl. Math. Mech. 28 (2007), pp. 27–35. doi: 10.1007/s10483-007-0104-x
   Web of Science ®Google Scholar
• M. Marion and J. Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal. 32 (1995), pp. 1170–1184. doi: 10.1137/0732054
   Web of Science ®Google Scholar
• S. Markus, E.N. Houstis, A.C. Catlin, J.R. Rice, P. Tsompanopoulou, E. Vavalis, D. Gottfried, K. Su, and G. Balakrishnan, An agent-based netcentric framework for multidisciplinary problem solving environments (MPSE), Internat. J. Comput. Engrg. Sci. 1 (2000), pp. 33–60. doi: 10.1142/S1465876300000045
   Google Scholar
• A. Márquez, S. Meddahi, and F.-J. Sayas, A decoupled preconditioning technique for a mixed Stokes–Darcy model, J. Sci. Comput. 57 (2013), pp. 174–192. doi: 10.1007/s10915-013-9700-5
   Web of Science ®Google Scholar
• M. Mu, Solving composite problems with interface relaxation, SIAM J. Sci. Comput. 20 (1999), pp. 1394–1416. doi: 10.1137/S1064827597321180
   Web of Science ®Google Scholar
• 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), pp. 1801–1813. doi: 10.1137/050637820
   Web of Science ®Google Scholar
• M. Mu and X.H. Zhu, Decoupled schemes for a non-stationary mixed Stokes–Darcy model, Math. Comput. 79 (2010), pp. 707–731. doi: 10.1090/S0025-5718-09-02302-3
   Web of Science ®Google Scholar
• P. Saffman, On the boundary condition at the surface of a porous media, Stud. Appl. Math. 50 (1971), pp. 93–101.
   Web of Science ®Google Scholar
• L. Shan and H.B. Zheng, Partitioned time stepping method for fully evolutionary Stokes–Darcy flow with the Beavers–Joseph interface conditions, SIAM J. Numer. Anal. 51 (2013), pp. 813–839. doi: 10.1137/110828095
   Web of Science ®Google Scholar
• L. Shan, H.B. Zheng, and W.J. Layton, A decoupling method with different subdomain time steps for the nonstationary Stokes–Darcy model, Numer. Methods Partial Differ. Eqns. 29 (2013), pp. 549–583. doi: 10.1002/num.21720
   Web of Science ®Google Scholar
• J.M. Urquiza, D. N'Dri, A. Garon, and M.C. Delfour, Coupling Stokes and Darcy equations, Appl. Numer. Math. 58 (2008), pp. 525–538. doi: 10.1016/j.apnum.2006.12.006
   Web of Science ®Google Scholar
• D. Vassilev, C.Q. Wang, and I. Yotov, Domain decomposition for coupled Stokes and Darcy flows, Comput. Methods Appl. Mech. Engrg. 268 (2014), pp. 264–283. doi: 10.1016/j.cma.2013.09.009
   Web of Science ®Google Scholar
• J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), pp. 231–237. doi: 10.1137/0915016
   Web of Science ®Google Scholar
• J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), pp. 1759–1777. doi: 10.1137/S0036142992232949
   Web of Science ®Google Scholar
• J.C. Xu and A.H. Zhou, Some local and parallel properties of finite element discretizations, Proceedings for Eleventh International Conference on Domain Decomposition Methods, Greenwich, 1999, pp. 140–147.
   Google Scholar
• J.C. Xu and A.H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comput. 69 (1999), pp. 881–910. doi: 10.1090/S0025-5718-99-01149-7
   Web of Science ®Google Scholar
• J.C. Xu and A.H. Zhou, Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math. 14 (2001), pp. 293–327. doi: 10.1023/A:1012284322811
   Web of Science ®Google Scholar
• T. Zhang and J.Y. Yuan, Two novel decoupling algorithms for the steady Stokes–Darcy model based on two-grid discretizations, Discrete Contin. Dyn. Syst. Ser. B. 19 (2014), pp. 849–865. doi: 10.3934/dcdsb.2014.19.849
   Web of Science ®Google Scholar
• L.Y. Zuo and Y.R. Hou, A decoupling two-grid algorithm for the mixed Stokes–Darcy model with the Beavers–Joseph interface condition, Numer. Methods Partial Differ. Eqns. 3 (2014), pp. 1066–1082. doi: 10.1002/num.21860
   Web of Science ®Google Scholar
• L.Y. Zuo and Y.R. Hou, A two-grid decoupling method for the mixed Stokes–Darcy model, J. Comput. Appl. Math. 275 (2015), pp. 139–147. doi: 10.1016/j.cam.2014.08.008
   Web of Science ®Google Scholar
• L.Y. Zuo and Y.R. Hou, Numerical analysis for the mixed Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition, Numer. Methods Partial Differ. Eqns. DOI:10.1002/num.21933.
   Web of Science ®Google Scholar