DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition

In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.

[1]  Andrea Toselli,et al.  Mixed hp-DGFEM for Incompressible Flows , 2002, SIAM J. Numer. Anal..

[2]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[3]  Susanne C. Brenner,et al.  Korn's inequalities for piecewise H1 vector fields , 2003, Math. Comput..

[4]  Mary F. Wheeler,et al.  Compatible algorithms for coupled flow and transport , 2004 .

[5]  A. Quarteroni,et al.  Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations , 2003 .

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

[7]  M. Fortin,et al.  A non‐conforming piecewise quadratic finite element on triangles , 1983 .

[8]  Béatrice Rivière.  Analysis of a Multi-Numerics/Multi-Physics Problem , 2004 .

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

[10]  A. Quarteroni,et al.  Mathematical analysis of the Navier-Stokes / Darcy coupling , 2006 .

[11]  R. Glowinski.  Finite element methods for incompressible viscous flow , 2003 .

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

[13]  Mary F. Wheeler,et al.  Numerical discretization of a Darcy–Forchheimer model , 2008, Numerische Mathematik.

[14]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[15]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[16]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[17]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

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

[19]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[20]  Christine Bernardi,et al.  Un résultat de trace pour les éléments finis de Crouzeix-Raviart, application à la discrétisation des équations de Darcy , 2007 .

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

[22]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[23]  Fédérale De Lausanne,et al.  DOMAIN DECOMPOSITION METHODS FOR THE COUPLING OF SURFACE AND GROUNDWATER FLOWS , 2004 .

[24]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[25]  Mary F. Wheeler,et al.  A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems , 2004, Math. Comput..

[26]  M. Crouzeix,et al.  Nonconforming finite elements for the Stokes problem , 1989 .

[27]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[28]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

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

[30]  B. Rivière,et al.  Esaim: Mathematical Modelling and Numerical Analysis a Splitting Method Using Discontinuous Galerkin for the Transient Incompressible Navier-stokes Equations , 2022 .