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.  Google Scholar

[2]

P. BochevC. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[5]

M. C. CaiM. 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.  Google Scholar

[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.  Google Scholar

[7]

M. C. CaiM. 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.  Google Scholar

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Y. CaoM. GunzburgerX. HuF. HuaX. 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.  Google Scholar

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Y. CaoM. GunzburgerF. 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.  Google Scholar

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Y. CaoM. GunzburgerX. 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.  Google Scholar

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Y. CaoM. GunzburgerX. 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.  Google Scholar

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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.  Google Scholar

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W. ChenM. GunzburgerF. 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.  Google Scholar

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M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph. D. dissertation, École Polytechnique Fédérale de Lausanne, 2004. Google Scholar

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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.  Google Scholar

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M. DiscacciatiE. 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.  Google Scholar

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M. DiscaaaiatiA. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[20]

R. GlowinskiT. 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.  Google Scholar

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N. HanspalA. WaghodeV. 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.  Google Scholar

[22]

X. HeJ. LiY. 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.  Google Scholar

[23]

F. Hecht, FreeFEM++, J. Numer. Math., 20 (2012), 251-265.   Google Scholar

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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.  Google Scholar

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F. Hua, Modeling, Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition, Ph. D. dissertation, The Florida State University, 2009.  Google Scholar

[26]

P. Z. HuangX. 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.  Google Scholar

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[28]

H. JiaH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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W. J. LaytonF. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2002), 2195-2218.  doi: 10.1137/S0036142901392766.  Google Scholar

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R. LiJ. LiZ. 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.  Google Scholar

[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.  Google Scholar

[36]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, 1972.  Google Scholar

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A. MarquezS. 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.  Google Scholar

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M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

G. PacquautJ. BruchonN. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[45]

L. ShanH. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[2]

P. BochevC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. C. CaiM. 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.  Google Scholar

[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.  Google Scholar

[7]

M. C. CaiM. 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.  Google Scholar

[8]

Y. CaoM. GunzburgerX. HuF. HuaX. 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.  Google Scholar

[9]

Y. CaoM. GunzburgerF. 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.  Google Scholar

[10]

Y. CaoM. GunzburgerX. 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.  Google Scholar

[11]

Y. CaoM. GunzburgerX. 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.  Google Scholar

[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.  Google Scholar

[13]

W. ChenM. GunzburgerF. 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.  Google Scholar

[14]

M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph. D. dissertation, École Polytechnique Fédérale de Lausanne, 2004. Google Scholar

[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.  Google Scholar

[16]

M. DiscacciatiE. 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.  Google Scholar

[17]

M. DiscaaaiatiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. GlowinskiT. 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.  Google Scholar

[21]

N. HanspalA. WaghodeV. 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.  Google Scholar

[22]

X. HeJ. LiY. 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.  Google Scholar

[23]

F. Hecht, FreeFEM++, J. Numer. Math., 20 (2012), 251-265.   Google Scholar

[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.  Google Scholar

[25]

F. Hua, Modeling, Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition, Ph. D. dissertation, The Florida State University, 2009.  Google Scholar

[26]

P. Z. HuangX. 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.  Google Scholar

[27]

T. J. R. HughesL. 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.  Google Scholar

[28]

H. JiaH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

W. J. LaytonF. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40 (2002), 2195-2218.  doi: 10.1137/S0036142901392766.  Google Scholar

[34]

R. LiJ. LiZ. 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.  Google Scholar

[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.  Google Scholar

[36]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, 1972.  Google Scholar

[37]

A. MarquezS. 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.  Google Scholar

[38]

M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

G. PacquautJ. BruchonN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

L. ShanH. 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.  Google Scholar

[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.  Google Scholar

[47]

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Figure 1.  The pressure line by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
Figure 2.  Streamlines for the numerical velocity by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
Figure 3.  The velocity streamlines of the backward facing step flow with two interface conditions: Case 1 (top), Case 2 (bottom)
Table 1.  The convergence performance and CPU time by SFEM
h$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac14$3.666e-1-2.051e-1-9.850e-1-0.101
$\frac{1}{16}$9.850e-20.9485.103e-21.0049.534e-21.6843.547
$\frac{1}{64}$2.476e-20.9961.280e-20.9983.858e-20.653214.951
h$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac14$3.666e-1-2.051e-1-9.850e-1-0.101
$\frac{1}{16}$9.850e-20.9485.103e-21.0049.534e-21.6843.547
$\frac{1}{64}$2.476e-20.9961.280e-20.9983.858e-20.653214.951
Table 2.  The convergence performance and CPU time by TGM
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac14$$\frac{1}{16}$9.577e-2-5.109e-2-1.741e-1-0.181
$\frac{1}{8}$$\frac{1}{64}$2.405e-20.9971.275e-21.0013.799e-21.0982.127
$\frac{1}{16}$$\frac{1}{256}$6.014e-31.0003.187e-31.0009.134e-31.02839.827
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac14$$\frac{1}{16}$9.577e-2-5.109e-2-1.741e-1-0.181
$\frac{1}{8}$$\frac{1}{64}$2.405e-20.9971.275e-21.0013.799e-21.0982.127
$\frac{1}{16}$$\frac{1}{256}$6.014e-31.0003.187e-31.0009.134e-31.02839.827
Table 3.  The convergence performance and CPU time by StbTGM
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\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-20.9971.375e-21.0003.682e-21.0321.351
$\frac{1}{16}$$\frac{1}{256}$6.012e-31.0003.437e-31.0009.267e-30.99526.805
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac{1}{4}$$\frac{1}{16}$9.577e-2-5.494e-2-1.540e-1-0.117
$\frac{1}{8}$$\frac{1}{64}$2.404e-20.9971.375e-21.0003.682e-21.0321.351
$\frac{1}{16}$$\frac{1}{256}$6.012e-31.0003.437e-31.0009.267e-30.99526.805
Table 4.  The convergence performance by StbTGM with fixed $H = 1/8$
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$Rate
$\frac18$$\frac{1}{16}$9.470e-25.489e-25.040e-2
$\frac18$$\frac{1}{64}$2.404e-20.9891.375e-20.9993.682e-20.226
$\frac18$$\frac{1}{256}$7.033e-30.8873.525e-30.9823.704e-2-0.004
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$Rate
$\frac18$$\frac{1}{16}$9.470e-25.489e-25.040e-2
$\frac18$$\frac{1}{64}$2.404e-20.9891.375e-20.9993.682e-20.226
$\frac18$$\frac{1}{256}$7.033e-30.8873.525e-30.9823.704e-2-0.004
Table 5.  The convergence performance and CPU time by TGM-$(P_1, P_1, P_1)$
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\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-20.9961.386e-20.9992.904e+6-1.351
$\frac{1}{16}$$\frac{1}{256}$null-null-null--
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$RateCPU
$\frac{1}{4}$$\frac{1}{16}$9.553e-2-5.539e-2-4.594e+7-0.117
$\frac{1}{8}$$\frac{1}{64}$2.403e-20.9961.386e-20.9992.904e+6-1.351
$\frac{1}{16}$$\frac{1}{256}$null-null-null--
Table 6.  The approximation errors by StbTGM for $k = 0.1$
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$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-20.9991.006e-21.0065.853e-30.939
$\frac{1}{16}$$\frac{1}{256}$3.243e-31.0002.509e-31.0011.536e-30.965
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$Rate
$\frac{1}{4}$$\frac{1}{16}$5.181e-2-4.057e-2-2.15254e-2-
$\frac{1}{8}$$\frac{1}{64}$1.297e-20.9991.006e-21.0065.853e-30.939
$\frac{1}{16}$$\frac{1}{256}$3.243e-31.0002.509e-31.0011.536e-30.965
Table 7.  The approximation errors by StbTGM for $k = 0.01$
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$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-21.0001.610e-21.0898.070e-30.956
$\frac{1}{16}$$\frac{1}{256}$3.372e-31.0013.785e-31.0442.081e-30.978
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$Rate
$\frac{1}{4}$$\frac{1}{16}$5.397e-2-7.288e-2-3.03543e-2-
$\frac{1}{8}$$\frac{1}{64}$1.350e-21.0001.610e-21.0898.070e-30.956
$\frac{1}{16}$$\frac{1}{256}$3.372e-31.0013.785e-31.0442.081e-30.978
Table 8.  The approximation errors by StbTGM for $k = 0.001$
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$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-20.9996.834e-21.35416.980e-31.000
$\frac{1}{16}$$\frac{1}{256}$3.423e-31.0031.229e-21.2381.755e-30.996
Hh$\|\varphi-\varphi^h\|_{1, \Omega_p}$Rate$\|u-u^h\|_{1, \Omega_f}$Rate$\|p-p^h\|_{0, \Omega_f}$Rate
$\frac{1}{4}$$\frac{1}{16}$5.490e-2-4.467e-1-2.793e-2-
$\frac{1}{8}$$\frac{1}{64}$1.375e-20.9996.834e-21.35416.980e-31.000
$\frac{1}{16}$$\frac{1}{256}$3.423e-31.0031.229e-21.2381.755e-30.996
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