# American Institute of Mathematical Sciences

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

 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.
Citation: Tong Zhang, Jinyun Yuan. Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations. Discrete and 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. [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. [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. [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. [5] G. Beavers and D. Josephn, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197-207. doi: 10.1017/S0022112067001375. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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