Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow

Authors:
P. Hansbo;M. Juntunen
Affiliations:
Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-41296 Göteborg, Sweden;Department of Engineering Physics and Mathematics, Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

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 [J. Nitsche, Uber ein Variationsprinzip zur Losung von Dirichlet-Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 36 (1971) 9-15]. 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.