Primal Discontinuous Galerkin Methods for Time-Dependent Coupled Surface and Subsurface Flow

This paper introduces and analyzes a numerical method based on discontinuous finite element methods for solving the two-dimensional coupled problem of time-dependent incompressible Navier-Stokes equations with the Darcy equations through Beaver-Joseph-Saffman’s condition on the interface. The proposed method employs Crank-Nicolson discretization in time (which requires one step of a first order scheme namely backward Euler) and primal DG method in space. With the correct assumption on the first time step optimal error estimates are obtained that are high order in space and second order in time.

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

[2]  Alfio Quarteroni,et al.  Robin-Robin Domain Decomposition Methods for the Stokes-Darcy Coupling , 2007, SIAM J. Numer. Anal..

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

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

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

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

[7]  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..

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

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

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

[11]  Béatrice Rivière,et al.  Analysis of time-dependent Navier–Stokes flow coupled with Darcy flow , 2008, J. Num. Math..

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

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

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

[15]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

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

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

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

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

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

[21]  B. Rivière,et al.  On the solution of the coupled Navier–Stokes and Darcy equations , 2009 .

[22]  Jinchao Xu,et al.  A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow , 2007, SIAM J. Numer. Anal..

[23]  R. Aris,et al.  FINITE ELEMENT FORMULATIONS FOR LARGE-SCALE, COUPLED FLOWS IN ADJACENT POROUS AND OPEN FLUID DOMAINS , 1994 .

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

[25]  M. Wheeler.  An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .

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

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

[28]  B. Rivière,et al.  Estimation of penalty parameters for symmetric interior penalty Galerkin methods , 2007 .