Abstract
The Brinkman equations describe the flow of a viscous fluid in a porous matrix. Mathematically the Brinkman model is a parameter-dependent combination of the Darcy and Stokes models. A dual mixed framework is introduced for the problem, and H(div)-conforming finite elements are used with Nitsche’s method to obtain a stable formulation. We show the formulation to be stable in a mesh-dependent norm for all values of the parameter and introduce a postprocessing scheme for the pressure, which gives optimal convergence for the pressure.
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References
T. Arbogast and H. L. Lehr. Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci., 10(3):291–302, 2006
F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, Berlin, 1991
P. Hansbo and M Juntunen. Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. Appl. Numer. Math., 59(6):1274–1289, 2009
M. Juntunen and R. Stenberg. Analysis of finite element methods for the Brinkman problem. Calcolo, 2009. doi:10.1007/s10092-009-0017-6
J. Könnö and R. Stenberg. Analysis of H(div)-conforming finite elements for the Brinkman problem. Helsinki University of Technology Institute of Mathematics Research Report A 582 (2010)
T. Lévy. Loi de Darcy ou loi de Brinkman? C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 292(12):871–874, Erratum (17):1239, 1981
C. Lovadina and R. Stenberg. Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp., 75(256):1659–1674 (electronic), 2006
J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9–15, 1971. Collection of articles dedicated to Lothar Collatz on his sixtieth birthday
J. Schöberl. Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Institute for Scientific Computing, Texas AM University, 2001
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Könnö, J., Stenberg, R. (2010). Non-Conforming Finite Element Method for the Brinkman Problem. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds) Numerical Mathematics and Advanced Applications 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11795-4_55
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DOI: https://doi.org/10.1007/978-3-642-11795-4_55
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