Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T12:53:34.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

Part of: Partial differential equations, boundary value problems Incompressible viscous fluids

Published online by Cambridge University Press:  27 January 2016

Jing Zhao  and
Tong Zhang
Jing Zhao*
Affiliation:
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, P.R. China and Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba 81531-980, P.R. Brazil
Tong Zhang
Affiliation:
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, P.R. China and Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba 81531-980, P.R. Brazil
*
*Corresponding author. Email addresses:[email protected] (J. Zhao), [email protected] (T. Zhang)
Get access

Abstract

Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the H1-norm for velocity and piezometric approximations and the L2-norm for pressure are established under mesh sizes satisfying h = H2. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.

Keywords

Navier-Stokes equationsDarcy's lawmultimodeling problemstwo-grid method

MSC classification

Secondary: 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76D07: Stokes and related (Oseen, etc.) flows
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 1 , February 2016 , pp. 60 - 79
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Badea, L., Discacciati, M. and Quarteroni, A., Numerical analysis of the Navier-Stokes/Darcy coupling, Num. Math. 115, 195–227 (2010).CrossRefGoogle Scholar
[2]Beavers, G. and Joseph, D., Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30, 197–207 (1967).Google Scholar
[3]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer, New York (1991).Google Scholar
[4]Cai, M.C., Mu, M. and Xu, J.C., Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal. 47, 3325–3338 (2009).Google Scholar
[5]Chidyagwai, E. and Riviere, B., On the solution of the coupled Navier-Stokes and Darcy equations, Comput. Methods Appl. Mech. Engrg. 198, 3806–3820 (2009).Google Scholar
[6]Dawson, C. and Wheeler, M., Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. Math. 180, 191–203 (1994).Google Scholar
[7]Dawson, C., Wheeler, M. and Woodward, C., A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal. 35, 435–452 (1998).Google Scholar
[8]Discacciati, M., Miglo, E. and Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math. 43, 57–74 (2002).CrossRefGoogle Scholar
[9]Discacciati, M. and Quarteroni, A., Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations, Comput. Vis. Sci. 6, 93–103 (2004).Google Scholar
[10]Discacciati, M., Domain decomposition methods for the coupling of surface and groundwater flows, Ph.D thesis, Ecole Polythchnique Federale de Lausamme, Lausanne, France (2004).Google Scholar
[11]Girault, V. and Riviere, B., DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM J. Numer. Anal. 47, 2052–2089 (2009).Google Scholar
[12]Jager, W. and Mikelic, A., On the interface boundary condition of Beavers, Joseph, and Saffma, SIAM J. Appl. Math. 60, 1111–1127 (2000).Google Scholar
[13]Layton, W., A two-level discretization method for the Navier-Stokes equations, Comput. Math. Appl. 26, 33–38 (1993).Google Scholar
[14]Layton, W. and Leferink, J., Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput. 69, 263–274 (1995).Google Scholar
[15]Layton, W., Schieweck, F. and Yotov, I., Coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 40, 2195–2218 (2003).Google Scholar
[16]Layton, W. and Tobiska, L., A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal. 35, 2035–2054 (1998).Google Scholar
[17]Li, J., He, Y.N. and Xu, H., A multi-level stabilized finite element method for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engng. 196, 2852–2862 (2007).Google Scholar
[18]He, Y.N. and Li, K.T., Two-level stabilized finite element methods for steady Navier-Stokes equations, Computing 75, 337–351 (2005).CrossRefGoogle Scholar
[19]He, Y.N., Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 41, 1263–1285 (2003).Google Scholar
[20]He, Y.N. and Wang, A.W., A simplified two-level method for the steady Navier-Stokes equations, Comput. Methods Appl. Mech. Engng. 197, 1568–1576 (2008).Google Scholar
[21]He, Y.N., Miao, H.L. and Ren, C.F., A two-lev el finite element Galerkin methodforthe nonstationary Navier-Stokes equations II: Time discretization, J. Comput. Math. 22, 33–54 (2004).Google Scholar
[22]Mu, M. and Xu, J.C., A two-grid method of a mixed stokes-darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 45, 1801–1813 (2007).Google Scholar
[23]Nield, D.A. and Bejan, A., Convection in Porous Media, Springer, New York (1999).CrossRefGoogle Scholar
[24]Saffman, P., On the boundary condition at the surface of a porous media, Stud. Appl. Math. 50, 93–101 (1971).Google Scholar
[25]Temam, R., Navier-Stokes Equations, North-Holland, Amsterdam (1979).Google Scholar
[26]Xu, J.C., A novel two-grid method for semi-linear elliptic equations, SIAM J. Sci. Comput. 15, 231–237 (1994).CrossRefGoogle Scholar
[27]Xu, J.C., Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33, 1759–1777 (1996).Google Scholar
[28]Zhang, T., The semidiscrete finite volume element method for nonlinear convection-diffusion problem, Appl. Math. Comput. 217, 7546–7556 (2011).Google Scholar
[29]Zhang, T., Two-grid characteristic finite volume methods for nonlinear parabolic problems, J. Comput. Math. 31, 470–487 (2013).Google Scholar
[30]Zhang, T. and Yuan, J.Y., Two novel decoupling finite element algorithms for the steady Stokes-Darcy model based on two grid discretization, Discrete Continuous Dynam. Systems-B 19, 849–865 (2014).Google Scholar
[31]Zhang, T., Zhong, H. and Zhao, J., A full discrete two-grid finite-volume method for a nonlinear parabolic problem, Int. J. Comput. Math. 88, 1644–1663 (2011).CrossRefGoogle Scholar