A strongly conservative finite element method for the coupling of Stokes and Darcy flow

We consider a model of coupled free and porous media flow governed by Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition. This model is discretized using divergence-conforming finite elements for the velocities in the whole domain. Discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. This discretization is strongly conservative in H^d^i^v(@W) and we show convergence. Numerical results validate our findings and indicate optimal convergence orders.

[1]  P. Hansbo,et al.  CHALMERS FINITE ELEMENT CENTER Preprint 2000-06 Discontinuous Galerkin Methods for Incompressible and Nearly Incompressible Elasticity by Nitsche ’ s Method , 2007 .

[2]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[3]  T. Bratanow.  Finite element approximations of the Navier-Stokes equations , 1981 .

[4]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[5]  P. Saffman.  On the Boundary Condition at the Surface of a Porous Medium , 1971 .

[6]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[7]  Ilio Galligani,et al.  Mathematical Aspects of Finite Element Methods , 1977 .

[8]  Guido Kanschat,et al.  A Note on Discontinuous Galerkin Divergence-free Solutions of the Navier–Stokes Equations , 2007, J. Sci. Comput..

[9]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[10]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[11]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[12]  Douglas N. Arnold,et al.  Approximation by quadrilateral finite elements , 2000, Math. Comput..

[13]  P. Hansbo,et al.  Stabilized Crouzeix‐Raviart element for the Darcy‐Stokes problem , 2005 .

[14]  Junping Wang,et al.  New Finite Element Methods in Computational Fluid Dynamics by H(div) Elements , 2007, SIAM J. Numer. Anal..

[15]  Willi Jäger,et al.  On the boundary conditions at the contact interface between a porous medium and a free fluid , 1996 .

[16]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[17]  Willi Jäger,et al.  Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization , 2009 .

[18]  E. Miglio,et al.  Mathematical and numerical models for coupling surface and groundwater flows , 2002 .

[19]  Ivan Yotov,et al.  Coupling Fluid Flow with Porous Media Flow , 2002, SIAM J. Numer. Anal..

[20]  G. Gatica,et al.  A conforming mixed finite-element method for the coupling of fluid flow with porous media flow , 2008 .

[21]  Vahid Nassehi,et al.  Modelling of combined Navier–Stokes and Darcy flows in crossflow membrane filtration , 1998 .

[22]  Béatrice Rivière,et al.  Locally Conservative Coupling of Stokes and Darcy Flows , 2005 .

[23]  T. Arbogast,et al.  A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium , 2007 .

[24]  Willi Jäger,et al.  On The Interface Boundary Condition of Beavers, Joseph, and Saffman , 2000, SIAM J. Appl. Math..

[25]  V. Nassehi,et al.  Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations , 2006 .

[26]  VIVETTE GIRAULT,et al.  DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition , 2009, SIAM J. Numer. Anal..

[27]  D. Joseph,et al.  Boundary conditions at a naturally permeable wall , 1967, Journal of Fluid Mechanics.

[28]  Béatrice Rivière,et al.  Analysis of a Discontinuous Finite Element Method for the Coupled Stokes and Darcy Problems , 2005, J. Sci. Comput..

[29]  P. Hansbo,et al.  A unified stabilized method for Stokes' and Darcy's equations , 2007 .