# American Institute of Mathematical Sciences

January 2019, 24(1): 387-402. doi: 10.3934/dcdsb.2018109

## Two-grid finite element method for the stabilization of mixed Stokes-Darcy model

 1 College of Science, Donghua University, Shanghai 201620, China 2 Department of Mathematics, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China 3 College of Science, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China 4 Department of Mathematics, University of Houston, Houston, TX 77024, USA 5 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

* Corresponding author: Feng Shi (shi.feng@hit.edu.cn)

Received  April 2017 Revised  August 2017 Published  March 2018

Fund Project: The first author is supported by NSFC (Grant Nos. 11501097 and 11471071). The second author is partially supported by NSFC (Grant Nos. 11201369 and 11771337). The third author is partially subsidized by Basic Research Program of Shenzhen (Grant No. JCYJ20150831112754988)

A two-grid discretization for the stabilized finite element method for mixed Stokes-Darcy problem is proposed and analyzed. The lowest equal-order velocity-pressure pairs are used due to their simplicity and attractive computational properties, such as much simpler data structures and less computer memory for meshes and algebraic system, easier interpolations, and convenient usages of many existing preconditioners and fast solvers in simulations, which make these pairs a much popular choice in engineering practice; see e.g., [4,27]. The decoupling methods are adopted for solving coupled systems based on the significant features that decoupling methods can allow us to solve the submodel problems independently by using most appropriate numerical techniques and preconditioners, and also to reduce substantial coding tasks. The main idea in this paper is that, on the coarse grid, we solve a stabilized finite element scheme for coupled Stokes-Darcy problem; then on the fine grid, we apply the coarse grid approximation to the interface conditions, and solve two independent subproblems: one is the stabilized finite element method for Stokes subproblem, and another one is the Darcy subproblem. Optimal error estimates are derived, and several numerical experiments are carried out to demonstrate the accuracy and efficiency of the two-grid stabilized finite element algorithm.

Citation: Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109
##### References:
 [1] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid. Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375. [2] P. Bochev, C. Dohrmann and M. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal., 44 (2006), 82-101. doi: 10.1137/S0036142905444482. [3] Y. Boubendir and S. Tlupova, Domain decomposition methods for solving Stokes-Darcy problems with bondary integrals, SIAM J. Sci. Comput., 35 (2013), B82-B106. doi: 10.1137/110838376. [4] A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8. [5] 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), 346-355. doi: 10.1016/j.cam.2009.07.029. [6] M. C. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math., 236 (2012), 2452-2465. doi: 10.1016/j.cam.2011.12.003. [7] 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), 3325-3338. doi: 10.1137/080721868. [8] Y. Cao, M. Gunzburger, X. Hu, F. Hua, X. Wang and W. Zhao, Finite element approximation for Stokes-Darcy flow with Beavers-Joseph interface conditions, SIAM J. Numer. Anal., 47 (2010), 4239-4256. doi: 10.1137/080731542. [9] Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Comm. Math. Sci., 8 (2010), 1-25. doi: 10.4310/CMS.2010.v8.n1.a2. [10] Y. Cao, M. Gunzburger, X. He and X. Wang, Robin-Robin domain decomposition methods for the steady Stokes-Darcy model with Beaver-Joseph interface condition, Numer. Math., 117 (2011), 601-629. doi: 10.1007/s00211-011-0361-8. [11] Y. Cao, M. Gunzburger, X. He and X. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems, Math. Comput., 83 (2014), 1617-1644. doi: 10.1090/S0025-5718-2014-02779-8. [12] Y. Cao, Y. Chu, X. He and M. Wei, Decoupling the stationary Navier-Stokes-Darcy system with the Beavers-Joseph-Saffman interface condition, Abstr. Appl. Anal. , 2013 (2013), Art. ID 136483, 10 pp. [13] 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), 1064-1084. doi: 10.1137/080740556. [14] M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph. D. dissertation, École Polytechnique Fédérale de Lausanne, 2004. [15] 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. Visual Sci., 6 (2004), 93-103. doi: 10.1007/s00791-003-0113-0. [16] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2002), 57-74. doi: 10.1016/S0168-9274(02)00125-3. [17] M. Discaaaiati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268. doi: 10.1137/06065091X. [18] 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 (eds. F. Brezzi et al), Springer, Milan, (2003), 3-20. [19] 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), 2052-2089. doi: 10.1137/070686081. [20] R. Glowinski, T. 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), 361-369. doi: 10.1016/S0764-4442(99)80376-0. [21] N. Hanspal, A. Waghode, V. Nassehi and R. Wakeman, Numerical analysis of coupled Stokes/Darcy flow in industrial filtrations, Transp. Porous Media, 64 (2006), 1573-1634. doi: 10.1007/s11242-005-1457-3. [22] 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 J. Sci. Comput., 37 (2015), S264-S290. doi: 10.1137/140965776. [23] F. Hecht, FreeFEM++, J. Numer. Math., 20 (2012), 251-265. [24] Y. R. Hou, Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes-Darcy model, Appl. Math. Letters, 57 (2016), 90-96. doi: 10.1016/j.aml.2016.01.007. [25] F. Hua, Modeling, Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition, Ph. D. dissertation, The Florida State University, 2009. [26] P. Z. Huang, X. L. Feng and H. Y. Su, Two-level defect-correction locally stabilized finite element method for the steady Navier-Stokese quations, Nonlinear Anal. Real World Appl., 14 (2013), 1171-1181. doi: 10.1016/j.nonrwa.2012.09.008. [27] T. J. R. Hughes, L. P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: Ⅴ. Circumventing the babuska-brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59 (1986), 85-99. doi: 10.1016/0045-7825(86)90025-3. [28] H. Jia, H. Jia and Y. Huang, A modified two-grid decoupling method for the mixed Navier-Stokes/Darcy Model, Comput. Math. Appl., 72 (2016), 1142-1152. doi: 10.1016/j.camwa.2016.06.033. [29] B. Jiang, A parallel domain decomposition method for coupling of surface and groundwarter flows, Comput. Methods Appl. Mech. Engrg., 198 (2009), 947-957. doi: 10.1016/j.cma.2008.11.001. [30] F. D. Kong and X. C. Cai, A highly scalable multilevel Schwarz method with boundary geometry preserving coarse spaces for 3D elasticity problems on domains with complex geometry, SIAM J. Sci. Comput., 38 (2016), C73-C95. doi: 10.1137/15M1010567. [31] F. D. Kong and X. C. Cai, Scalability study of an implicit solver for coupled fluid-structure interaction problems on unstructured meshes in 3D, Int. J. High Perform. Comput. Appl., 32 (2018), 207-219. doi: 10.1177/1094342016646437. [32] F. D. Kong and X. C. Cai, A scalable nonlinear fluid-structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D." Journal of Computational Physics, J. Comput. Phys., 340 (2017), 498-518. doi: 10.1016/j.jcp.2017.03.043. [33] W. J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2002), 2195-2218. doi: 10.1137/S0036142901392766. [34] R. Li, J. Li, Z. X. Chen and Y. L. Gao, A stabilized finite element method based on two local Gauss integrations for a coupled Stokes-Darcy problem, J. Comput. Appl. Math., 292 (2016), 92-104. doi: 10.1016/j.cam.2015.06.014. [35] J. Li and Y. N. He, A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214 (2008), 58-65. doi: 10.1016/j.cam.2007.02.015. [36] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, 1972. [37] A. Marquez, S. Meddahi and F. J. Sayas, A decoupled preconditioning technique for a mixed Stokes-Darcy model, J. Sci. Comput., 57 (2013), 174-192. doi: 10.1007/s10915-013-9700-5. [38] M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731. [39] 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), 1801-1813. doi: 10.1137/050637820. [40] K. Nafa, Equal order approximations enriched with bubbles for coupled Stokes-Darcy problem, J. Comput. Appl. Math., 270 (2014), 275-282. doi: 10.1016/j.cam.2014.01.010. [41] K. Nafa, Stability of some low-order approximations for Stokes problem, Internat. J. Numer. Methods Fluids, 56 (2008), 753-765. doi: 10.1002/fld.1553. [42] G. Pacquaut, J. Bruchon, N. Moulin and S. Drapier, Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes-Darcy flows, Internat. J. Numer. Methods Fluids, 69 (2012), 459-480. doi: 10.1002/fld.2569. [43] 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), 2692-2699. doi: 10.1016/j.cma.2009.03.011. [44] P. Saffman, On the boundary condition at the surface of a porous media, Stud. Appl. Math., 50 (1971), 93-101. doi: 10.1002/sapm197150293. [45] 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), 549-583. doi: 10.1002/num.21720. [46] 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), 813-839. doi: 10.1137/110828095. [47] 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), 849-865. doi: 10.3934/dcdsb.2014.19.849. [48] H. B. Zheng, Y. R. Hou and F. Shi, A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow, SIAM J. Sci. Comput., 32 (2010), 1346-1360. doi: 10.1137/090771508. [49] 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., 30 (2014), 1066-1082. doi: 10.1002/num.21860.

show all references

##### References:
 [1] G. Beavers and D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid. Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375. [2] P. Bochev, C. Dohrmann and M. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal., 44 (2006), 82-101. doi: 10.1137/S0036142905444482. [3] Y. Boubendir and S. Tlupova, Domain decomposition methods for solving Stokes-Darcy problems with bondary integrals, SIAM J. Sci. Comput., 35 (2013), B82-B106. doi: 10.1137/110838376. [4] A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8. [5] 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), 346-355. doi: 10.1016/j.cam.2009.07.029. [6] M. C. Cai and M. Mu, A multilevel decoupled method for a mixed Stokes/Darcy model, J. Comput. Appl. Math., 236 (2012), 2452-2465. doi: 10.1016/j.cam.2011.12.003. [7] 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), 3325-3338. doi: 10.1137/080721868. [8] Y. Cao, M. Gunzburger, X. Hu, F. Hua, X. Wang and W. Zhao, Finite element approximation for Stokes-Darcy flow with Beavers-Joseph interface conditions, SIAM J. Numer. Anal., 47 (2010), 4239-4256. doi: 10.1137/080731542. [9] Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Comm. Math. Sci., 8 (2010), 1-25. doi: 10.4310/CMS.2010.v8.n1.a2. [10] Y. Cao, M. Gunzburger, X. He and X. Wang, Robin-Robin domain decomposition methods for the steady Stokes-Darcy model with Beaver-Joseph interface condition, Numer. Math., 117 (2011), 601-629. doi: 10.1007/s00211-011-0361-8. [11] Y. Cao, M. Gunzburger, X. He and X. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems, Math. Comput., 83 (2014), 1617-1644. doi: 10.1090/S0025-5718-2014-02779-8. [12] Y. Cao, Y. Chu, X. He and M. Wei, Decoupling the stationary Navier-Stokes-Darcy system with the Beavers-Joseph-Saffman interface condition, Abstr. Appl. Anal. , 2013 (2013), Art. ID 136483, 10 pp. [13] 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), 1064-1084. doi: 10.1137/080740556. [14] M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph. D. dissertation, École Polytechnique Fédérale de Lausanne, 2004. [15] 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. Visual Sci., 6 (2004), 93-103. doi: 10.1007/s00791-003-0113-0. [16] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2002), 57-74. doi: 10.1016/S0168-9274(02)00125-3. [17] M. Discaaaiati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268. doi: 10.1137/06065091X. [18] 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 (eds. F. Brezzi et al), Springer, Milan, (2003), 3-20. [19] 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), 2052-2089. doi: 10.1137/070686081. [20] R. Glowinski, T. 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), 361-369. doi: 10.1016/S0764-4442(99)80376-0. [21] N. Hanspal, A. Waghode, V. Nassehi and R. Wakeman, Numerical analysis of coupled Stokes/Darcy flow in industrial filtrations, Transp. Porous Media, 64 (2006), 1573-1634. doi: 10.1007/s11242-005-1457-3. [22] 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 J. Sci. Comput., 37 (2015), S264-S290. doi: 10.1137/140965776. [23] F. Hecht, FreeFEM++, J. Numer. Math., 20 (2012), 251-265. [24] Y. R. Hou, Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes-Darcy model, Appl. Math. Letters, 57 (2016), 90-96. doi: 10.1016/j.aml.2016.01.007. [25] F. Hua, Modeling, Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition, Ph. D. dissertation, The Florida State University, 2009. [26] P. Z. Huang, X. L. Feng and H. Y. Su, Two-level defect-correction locally stabilized finite element method for the steady Navier-Stokese quations, Nonlinear Anal. Real World Appl., 14 (2013), 1171-1181. doi: 10.1016/j.nonrwa.2012.09.008. [27] T. J. R. Hughes, L. P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: Ⅴ. Circumventing the babuska-brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59 (1986), 85-99. doi: 10.1016/0045-7825(86)90025-3. [28] H. Jia, H. Jia and Y. Huang, A modified two-grid decoupling method for the mixed Navier-Stokes/Darcy Model, Comput. Math. Appl., 72 (2016), 1142-1152. doi: 10.1016/j.camwa.2016.06.033. [29] B. Jiang, A parallel domain decomposition method for coupling of surface and groundwarter flows, Comput. Methods Appl. Mech. Engrg., 198 (2009), 947-957. doi: 10.1016/j.cma.2008.11.001. [30] F. D. Kong and X. C. Cai, A highly scalable multilevel Schwarz method with boundary geometry preserving coarse spaces for 3D elasticity problems on domains with complex geometry, SIAM J. Sci. Comput., 38 (2016), C73-C95. doi: 10.1137/15M1010567. [31] F. D. Kong and X. C. Cai, Scalability study of an implicit solver for coupled fluid-structure interaction problems on unstructured meshes in 3D, Int. J. High Perform. Comput. Appl., 32 (2018), 207-219. doi: 10.1177/1094342016646437. [32] F. D. Kong and X. C. Cai, A scalable nonlinear fluid-structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D." Journal of Computational Physics, J. Comput. Phys., 340 (2017), 498-518. doi: 10.1016/j.jcp.2017.03.043. [33] W. J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2002), 2195-2218. doi: 10.1137/S0036142901392766. [34] R. Li, J. Li, Z. X. Chen and Y. L. Gao, A stabilized finite element method based on two local Gauss integrations for a coupled Stokes-Darcy problem, J. Comput. Appl. Math., 292 (2016), 92-104. doi: 10.1016/j.cam.2015.06.014. [35] J. Li and Y. N. He, A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214 (2008), 58-65. doi: 10.1016/j.cam.2007.02.015. [36] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, 1972. [37] A. Marquez, S. Meddahi and F. J. Sayas, A decoupled preconditioning technique for a mixed Stokes-Darcy model, J. Sci. Comput., 57 (2013), 174-192. doi: 10.1007/s10915-013-9700-5. [38] M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731. [39] 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), 1801-1813. doi: 10.1137/050637820. [40] K. Nafa, Equal order approximations enriched with bubbles for coupled Stokes-Darcy problem, J. Comput. Appl. Math., 270 (2014), 275-282. doi: 10.1016/j.cam.2014.01.010. [41] K. Nafa, Stability of some low-order approximations for Stokes problem, Internat. J. Numer. Methods Fluids, 56 (2008), 753-765. doi: 10.1002/fld.1553. [42] G. Pacquaut, J. Bruchon, N. Moulin and S. Drapier, Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes-Darcy flows, Internat. J. Numer. Methods Fluids, 69 (2012), 459-480. doi: 10.1002/fld.2569. [43] 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), 2692-2699. doi: 10.1016/j.cma.2009.03.011. [44] P. Saffman, On the boundary condition at the surface of a porous media, Stud. Appl. Math., 50 (1971), 93-101. doi: 10.1002/sapm197150293. [45] 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), 549-583. doi: 10.1002/num.21720. [46] 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), 813-839. doi: 10.1137/110828095. [47] 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), 849-865. doi: 10.3934/dcdsb.2014.19.849. [48] H. B. Zheng, Y. R. Hou and F. Shi, A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow, SIAM J. Sci. Comput., 32 (2010), 1346-1360. doi: 10.1137/090771508. [49] 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., 30 (2014), 1066-1082. doi: 10.1002/num.21860.
The pressure line by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
Streamlines for the numerical velocity by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
The velocity streamlines of the backward facing step flow with two interface conditions: Case 1 (top), Case 2 (bottom)
The convergence performance and CPU time by SFEM
 h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac14$ 3.666e-1 - 2.051e-1 - 9.850e-1 - 0.101 $\frac{1}{16}$ 9.850e-2 0.948 5.103e-2 1.004 9.534e-2 1.684 3.547 $\frac{1}{64}$ 2.476e-2 0.996 1.280e-2 0.998 3.858e-2 0.653 214.951
 h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac14$ 3.666e-1 - 2.051e-1 - 9.850e-1 - 0.101 $\frac{1}{16}$ 9.850e-2 0.948 5.103e-2 1.004 9.534e-2 1.684 3.547 $\frac{1}{64}$ 2.476e-2 0.996 1.280e-2 0.998 3.858e-2 0.653 214.951
The convergence performance and CPU time by TGM
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac14$ $\frac{1}{16}$ 9.577e-2 - 5.109e-2 - 1.741e-1 - 0.181 $\frac{1}{8}$ $\frac{1}{64}$ 2.405e-2 0.997 1.275e-2 1.001 3.799e-2 1.098 2.127 $\frac{1}{16}$ $\frac{1}{256}$ 6.014e-3 1.000 3.187e-3 1.000 9.134e-3 1.028 39.827
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac14$ $\frac{1}{16}$ 9.577e-2 - 5.109e-2 - 1.741e-1 - 0.181 $\frac{1}{8}$ $\frac{1}{64}$ 2.405e-2 0.997 1.275e-2 1.001 3.799e-2 1.098 2.127 $\frac{1}{16}$ $\frac{1}{256}$ 6.014e-3 1.000 3.187e-3 1.000 9.134e-3 1.028 39.827
The convergence performance and CPU time by StbTGM
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac{1}{4}$ $\frac{1}{16}$ 9.577e-2 - 5.494e-2 - 1.540e-1 - 0.117 $\frac{1}{8}$ $\frac{1}{64}$ 2.404e-2 0.997 1.375e-2 1.000 3.682e-2 1.032 1.351 $\frac{1}{16}$ $\frac{1}{256}$ 6.012e-3 1.000 3.437e-3 1.000 9.267e-3 0.995 26.805
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac{1}{4}$ $\frac{1}{16}$ 9.577e-2 - 5.494e-2 - 1.540e-1 - 0.117 $\frac{1}{8}$ $\frac{1}{64}$ 2.404e-2 0.997 1.375e-2 1.000 3.682e-2 1.032 1.351 $\frac{1}{16}$ $\frac{1}{256}$ 6.012e-3 1.000 3.437e-3 1.000 9.267e-3 0.995 26.805
The convergence performance by StbTGM with fixed $H = 1/8$
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac18$ $\frac{1}{16}$ 9.470e-2 5.489e-2 5.040e-2 $\frac18$ $\frac{1}{64}$ 2.404e-2 0.989 1.375e-2 0.999 3.682e-2 0.226 $\frac18$ $\frac{1}{256}$ 7.033e-3 0.887 3.525e-3 0.982 3.704e-2 -0.004
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac18$ $\frac{1}{16}$ 9.470e-2 5.489e-2 5.040e-2 $\frac18$ $\frac{1}{64}$ 2.404e-2 0.989 1.375e-2 0.999 3.682e-2 0.226 $\frac18$ $\frac{1}{256}$ 7.033e-3 0.887 3.525e-3 0.982 3.704e-2 -0.004
The convergence performance and CPU time by TGM-$(P_1, P_1, P_1)$
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac{1}{4}$ $\frac{1}{16}$ 9.553e-2 - 5.539e-2 - 4.594e+7 - 0.117 $\frac{1}{8}$ $\frac{1}{64}$ 2.403e-2 0.996 1.386e-2 0.999 2.904e+6 - 1.351 $\frac{1}{16}$ $\frac{1}{256}$ null - null - null - -
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate CPU $\frac{1}{4}$ $\frac{1}{16}$ 9.553e-2 - 5.539e-2 - 4.594e+7 - 0.117 $\frac{1}{8}$ $\frac{1}{64}$ 2.403e-2 0.996 1.386e-2 0.999 2.904e+6 - 1.351 $\frac{1}{16}$ $\frac{1}{256}$ null - null - null - -
The approximation errors by StbTGM for $k = 0.1$
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.181e-2 - 4.057e-2 - 2.15254e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.297e-2 0.999 1.006e-2 1.006 5.853e-3 0.939 $\frac{1}{16}$ $\frac{1}{256}$ 3.243e-3 1.000 2.509e-3 1.001 1.536e-3 0.965
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.181e-2 - 4.057e-2 - 2.15254e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.297e-2 0.999 1.006e-2 1.006 5.853e-3 0.939 $\frac{1}{16}$ $\frac{1}{256}$ 3.243e-3 1.000 2.509e-3 1.001 1.536e-3 0.965
The approximation errors by StbTGM for $k = 0.01$
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.397e-2 - 7.288e-2 - 3.03543e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.350e-2 1.000 1.610e-2 1.089 8.070e-3 0.956 $\frac{1}{16}$ $\frac{1}{256}$ 3.372e-3 1.001 3.785e-3 1.044 2.081e-3 0.978
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.397e-2 - 7.288e-2 - 3.03543e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.350e-2 1.000 1.610e-2 1.089 8.070e-3 0.956 $\frac{1}{16}$ $\frac{1}{256}$ 3.372e-3 1.001 3.785e-3 1.044 2.081e-3 0.978
The approximation errors by StbTGM for $k = 0.001$
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.490e-2 - 4.467e-1 - 2.793e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.375e-2 0.999 6.834e-2 1.3541 6.980e-3 1.000 $\frac{1}{16}$ $\frac{1}{256}$ 3.423e-3 1.003 1.229e-2 1.238 1.755e-3 0.996
 H h $\|\varphi-\varphi^h\|_{1, \Omega_p}$ Rate $\|u-u^h\|_{1, \Omega_f}$ Rate $\|p-p^h\|_{0, \Omega_f}$ Rate $\frac{1}{4}$ $\frac{1}{16}$ 5.490e-2 - 4.467e-1 - 2.793e-2 - $\frac{1}{8}$ $\frac{1}{64}$ 1.375e-2 0.999 6.834e-2 1.3541 6.980e-3 1.000 $\frac{1}{16}$ $\frac{1}{256}$ 3.423e-3 1.003 1.229e-2 1.238 1.755e-3 0.996
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