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May  2014, 19(3): 849-865. doi: 10.3934/dcdsb.2014.19.849

Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations

Tong Zhang 1, and  Jinyun Yuan 2,

1. 

School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, China
2. 

Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba 81531-980, Brazil

Received  September 2013 Revised  November 2013 Published  February 2014

In this work, two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations are proposed and analyzed. Optimal error estimates for these variables are presented. Two grid decoupled scheme proposed by Mu and Xu (2007) is used to develop the two novel decoupling algorithms. For Algorithm 3.2, the optimal error estimates are obtained for both ${\bf{u}}_f,\ p_f$ and $\phi$ with mesh sizes satisfying $H=\sqrt{h}$. For Algorithm 3.3, the convergence of $\phi$ in $H^1$-norm is improved form $H^2$ to $H^\frac{5}{2}$. Furthermore, the existing results in [17] are improved and supplemented. Finally, some numerical experiments are provided to show the efficiency and effectiveness of the developed algorithms.
Keywords: Stokes equations, Darcy's law, multimodeling problems, two grid method..
Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 76D0.
Citation: Tong Zhang, Jinyun Yuan. Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 849-865. doi: 10.3934/dcdsb.2014.19.849
References:
[1]

T. Arbogast and D. S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, Comput. Geosci., 11 (2007), 207-218. doi: 10.1007/s10596-007-9043-0.  Google Scholar

[2]

O. Axelsson and I. E. Kaporin, Minimum residual adaptive multilevel finite element procedure for the solution of nonlinear stationary problems, SIAM J. Numer. Anal., 35 (1998), 1213-1229. doi: 10.1137/S0036142995286428.  Google Scholar

[3]

O. Axelsson and W. Layton, A two-level method for the discretization of nonlinear boundary value problems, SIAM J. Numer. Anal., 33 (1996), 2359-2374. doi: 10.1137/S0036142993247104.  Google Scholar

[4]

O. Axelsson and A. Padiy, On a two level Newton type procedure applied for solving nonlinear elasticity problems, Internat. J. Numer. Methods Engrg., 49 (2000), 1479-1493. doi: 10.1002/1097-0207(20001230)49:12<1479::AID-NME4>3.0.CO;2-4.  Google Scholar

[5]

G. Beavers and D. Josephn, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375.  Google Scholar

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[7]

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

[8]

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), 93-103. doi: 10.1007/s00791-003-0113-0.  Google Scholar

[9]

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.  Google Scholar

[10]

W. Jager and A. Mikelic, On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 403-465.  Google Scholar

[11]

Y. N. He and K. Liu, A Multi-level Finite element method in space-time for the Navier-Stokes equations, Numer. Methods Partial Differential Eq., 21 (2005), 1052-1078. doi: 10.1002/num.20077.  Google Scholar

[12]

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), 1263-1285. doi: 10.1137/S0036142901385659.  Google Scholar

[13]

W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput., 69 (1995), 263-274. doi: 10.1016/0096-3003(94)00134-P.  Google Scholar

[14]

W. Layton, A. Meir and P. Schmidt, A two-level discretization method for the stationary MHD equations, Electron. Trans. Numer. Anal., 6 (1997), 198-210.  Google Scholar

[15]

W. Layton and L. Tobiska, A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal., 35 (1998), 2035-2054. doi: 10.1137/S003614299630230X.  Google Scholar

[16]

W. 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.  Google Scholar

[17]

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

[18]

J. C. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237. doi: 10.1137/0915016.  Google Scholar

[19]

J. C. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777. doi: 10.1137/S0036142992232949.  Google Scholar

[20]

T. Zhang, Two-grid characteristic finite volume methods for nonlinear parabolic problems, J. Comput. Math., 31 (2013), 470-487. doi: 10.4208/jcm.1304-m4288.  Google Scholar

[21]

T. Zhang and S. W. Xu, Two-level stabilized finite volume methods for the stationary Navier-Stokes equations, Adv. Appl. Math. Mech., 5 (2013), 19-35.  Google Scholar

show all references

References:
[1]

T. Arbogast and D. S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, Comput. Geosci., 11 (2007), 207-218. doi: 10.1007/s10596-007-9043-0.  Google Scholar

[2]

O. Axelsson and I. E. Kaporin, Minimum residual adaptive multilevel finite element procedure for the solution of nonlinear stationary problems, SIAM J. Numer. Anal., 35 (1998), 1213-1229. doi: 10.1137/S0036142995286428.  Google Scholar

[3]

O. Axelsson and W. Layton, A two-level method for the discretization of nonlinear boundary value problems, SIAM J. Numer. Anal., 33 (1996), 2359-2374. doi: 10.1137/S0036142993247104.  Google Scholar

[4]

O. Axelsson and A. Padiy, On a two level Newton type procedure applied for solving nonlinear elasticity problems, Internat. J. Numer. Methods Engrg., 49 (2000), 1479-1493. doi: 10.1002/1097-0207(20001230)49:12<1479::AID-NME4>3.0.CO;2-4.  Google Scholar

[5]

G. Beavers and D. Josephn, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375.  Google Scholar

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[7]

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

[8]

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), 93-103. doi: 10.1007/s00791-003-0113-0.  Google Scholar

[9]

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.  Google Scholar

[10]

W. Jager and A. Mikelic, On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23 (1996), 403-465.  Google Scholar

[11]

Y. N. He and K. Liu, A Multi-level Finite element method in space-time for the Navier-Stokes equations, Numer. Methods Partial Differential Eq., 21 (2005), 1052-1078. doi: 10.1002/num.20077.  Google Scholar

[12]

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), 1263-1285. doi: 10.1137/S0036142901385659.  Google Scholar

[13]

W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput., 69 (1995), 263-274. doi: 10.1016/0096-3003(94)00134-P.  Google Scholar

[14]

W. Layton, A. Meir and P. Schmidt, A two-level discretization method for the stationary MHD equations, Electron. Trans. Numer. Anal., 6 (1997), 198-210.  Google Scholar

[15]

W. Layton and L. Tobiska, A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal., 35 (1998), 2035-2054. doi: 10.1137/S003614299630230X.  Google Scholar

[16]

W. 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.  Google Scholar

[17]

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

[18]

J. C. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237. doi: 10.1137/0915016.  Google Scholar

[19]

J. C. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777. doi: 10.1137/S0036142992232949.  Google Scholar

[20]

T. Zhang, Two-grid characteristic finite volume methods for nonlinear parabolic problems, J. Comput. Math., 31 (2013), 470-487. doi: 10.4208/jcm.1304-m4288.  Google Scholar

[21]

T. Zhang and S. W. Xu, Two-level stabilized finite volume methods for the stationary Navier-Stokes equations, Adv. Appl. Math. Mech., 5 (2013), 19-35.  Google Scholar

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