×

Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. (English) Zbl 1162.76056

Summary: We use low order approximations, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Brinkman’s equation of porous media flow, applying an edge stabilization term to avoid locking. In order to handle the limiting case of Darcy flow, when only the velocity component normal to the boundary can be prescribed, we impose the boundary conditions weakly using Nitsche’s method. We show that this leads to a stable method for all choices of material parameters. Finally, we present some numerical examples verifying the theoretical predictions and showing the effect of the weak imposition of boundary conditions.

MSC:

76S05 Flows in porous media; filtration; seepage
76M10 Finite element methods applied to problems in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bazilevs, Y.; Hughes, T.J.R., Weak imposition of Dirichlet boundary conditions in fluid mechanics, Computers & fluids, 36, 1, 12-26, (2007) · Zbl 1115.76040
[2] Brenner, S.; Scott, L., The mathematical theory of finite element methods, (1994), Springer New York · Zbl 0804.65101
[3] Burman, E., A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty, SIAM journal on numerical analysis, 43, 5, 2012-2033, (2005) · Zbl 1111.65102
[4] Burman, E., Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem, Numerical methods for partial differential equations, 24, 1, 127-143, (2008) · Zbl 1139.76029
[5] Burman, E.; Fernández, M.A.; Hansbo, P., Continuous interior penalty finite element method for Oseen’s equations, SIAM journal on numerical analysis, 44, 3, 1248-1274, (2006) · Zbl 1344.76049
[6] Burman, E.; Hansbo, P., A unified stabilized method for Stokes’ and Darcy’s equations, Journal of computational and applied mathematics, 198, 1, 35-51, (2007) · Zbl 1101.76032
[7] Freund, J.; Stenberg, R., On weakly imposed boundary conditions for second order problems, (), 327-336
[8] Hughes, T.J.R.; Franca, L., A new finite element formulation for CFD: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Computer methods in applied mechanics and engineering, 65, 1, 85-96, (1987) · Zbl 0635.76067
[9] Mardal, K.; Tai, X.; Winther, R., A robust finite element method for darcy – stokes flow, SIAM journal on numerical analysis, 58, 5, 1605-1631, (2002) · Zbl 1037.65120
[10] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abhandlungen aus dem mathematischen seminar der universität Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[11] Thomée, V., Galerkin finite element methods for parabolic problems, (1997), Springer Berlin · Zbl 0884.65097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.