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 & 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.  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.  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.  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.  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.  doi: 10.1017/S0022112067001375.  Google Scholar

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (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.  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.  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.  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.   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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  doi: 10.1017/S0022112067001375.  Google Scholar

[6]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (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.  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.  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.  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.   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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.   Google Scholar

[1]

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

[2]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020184

[3]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196

[4]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, , () : -. doi: 10.3934/era.2020095

[5]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020076

[6]

Sondes khabthani, Lassaad Elasmi, François Feuillebois. Perturbation solution of the coupled Stokes-Darcy problem. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 971-990. doi: 10.3934/dcdsb.2011.15.971

[7]

Hao Li, Hai Bi, Yidu Yang. The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1775-1789. doi: 10.3934/dcdsb.2020002

[8]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

[9]

Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143

[10]

Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047

[11]

Jinchao Xu. The single-grid multilevel method and its applications. Inverse Problems & Imaging, 2013, 7 (3) : 987-1005. doi: 10.3934/ipi.2013.7.987

[12]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

[13]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006

[14]

Frédéric Sur, Michel Grédiac. Towards deconvolution to enhance the grid method for in-plane strain measurement. Inverse Problems & Imaging, 2014, 8 (1) : 259-291. doi: 10.3934/ipi.2014.8.259

[15]

Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065

[16]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495

[17]

Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369

[18]

Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497

[19]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17

[20]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]