Unified finite element discretizations of coupled Darcy-Stokes flow. (English) Zbl 1157.76026

Karper, Trygve; Mardal, Kent-Andre; Winther, Ragnar

Numer. Methods Partial Differ. Equations 25, No. 2, 311-326 (2009).

Summary: We discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy’s law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor-Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions is straightforward.

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
76S05	Flows in porous media; filtration; seepage
76D07	Stokes and related (Oseen, etc.) flows

Keywords:

nonconforming finite elements; interface conditions

References:

[1]	Gartling, Simulation of coupled viscous and porous flow problems, Comp Fluid Dyn 7 pp 23– (1996) · Zbl 0879.76104
[2]	Salinger, Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains, Int J Numer Methods Fluids 18 pp 1185– (1994) · Zbl 0807.76039
[3]	Layton, Coupling fluid flow with porous media flow, SIAM J Numer Anal 40 pp 2195– (2003) · Zbl 1037.76014
[4]	Miglio, Mathematical and numerical models for coupling surface and groundwater flows, Appl Numer Math 43 pp 57– (2002) · Zbl 1023.76048
[5]	Gatica, A conforming mixed finite element method for the coupling of fluid flow with porous media flow, Preprint 06-01, Departamento de Ingeniería Mathemática (2006)
[6]	Rivière, Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problem, J Sci Comput 22 pp 479– (2005)
[7]	Rivière, Locally conservative coupling of Stokes and Darcy flows, SIAM J Numer Anal 42 pp 1959– (2005) · Zbl 1084.35063
[8]	Arbogast, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, ICES Report 03-47 (2003)
[9]	Fortin, Old and new finite elements for incompressible flows, Int J Numer Methods Fluids 1 pp 347– (1981) · Zbl 0467.76030
[10]	Arbogast, A family of rectangular mixed elements with a continuous flux for second order elliptic problems, SIAM J Numer Anal 42 pp 1914– (2005) · Zbl 1081.65106
[11]	Beavers, Boundary conditions at a natural permeable wall, J Fluid Mech 30 pp 197– (1967)
[12]	Saffmann, On the boundary conditions at the interface of a porous medium, Studies Appl Math 1 pp 93– (1973)
[13]	Mardal, A robust finite element method for the Darcy-Stokes flow, SIAM J Numer Anal 40 pp 1605– (2002) · Zbl 1037.65120
[14]	Tai, A discrete de Rahm complex with enhanced smoothness, Calcolo 43 pp 287– (2006)
[15]	Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Numer Anal 8 pp 129– (1974) · Zbl 0338.90047
[16]	Brenner, Korn’s inequalities for picewise H1 vector fields, Math Comput 73 pp 1067– (2003)
[17]	Girault, Finite element methods for Navier Stokes equations (1986) · Zbl 0585.65077
[18]	Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions, SIAM J Numer Anal 41 pp 306– (2003)
[19]	Mardal, An observation on Korn’s inequality for nonconforming finite element methods, Math Comput 75 pp 1– (2006) · Zbl 1086.65112
[20]	Arnold, A stable finite element for the Stokes equation, Calcolo 21 pp 337– (1984)
[21]	Braess, Finite elements-Fast solvers and applications in solid mechanics (2001) · Zbl 0976.65099
[22]	Clement, Approximation by finite element functions using local regularizations, RAIRO Numer Anal 9 pp 77– (1975) · Zbl 0368.65008
[23]	Brezzi, Mixed and hybrid finite element methods (1991) · Zbl 0788.73002