Numerical computations with \(H(\mathop{div})\)-finite elements for the Brinkman problem. (English) Zbl 1348.76100

Summary: The \(H(\mathop{div})\)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in the authors’ article [Math. Models Methods Appl. Sci. 21, No. 11, 2227–2248 (2011; Zbl 1331.76115)]. Furthermore, the results are extended to cover a non-constant permeability. A hybridization technique for the problem is presented, complete with a convergence analysis and numerical verification. Finally, the numerical convergence studies are complemented with numerical examples of applications to domain decomposition and adaptive mesh refinement.


76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs


Zbl 1331.76115
Full Text: DOI arXiv


[1] Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1990) · Zbl 0724.76020
[2] Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113(3), 261–298 (1990) · Zbl 0724.76021
[3] Arbogast, T., Lehr, H.L.: Homogenization of a Darcy–Stokes system modeling vuggy porous media. Comput. Geosci. 10(3), 291–302 (2006) · Zbl 1197.76122
[4] Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1985) (1984) · Zbl 0593.76039
[5] Fraejis de Veubeke B.M.: Displacement and equilibrium models in the finite element method. In: Zienkiewics, O.C., Holister, G.S. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1965)
[6] Braess, D.: Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, New York (1992, 1997) · Zbl 0754.65084
[7] Brezzi, F., Douglas, J., Jr., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51(2), 237–250 (1987) · Zbl 0631.65107
[8] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) · Zbl 0788.73002
[9] Burman, E., Hansbo, P.: Stabilized Crouzeix–Raviart element for the Darcy–Stokes problem. Numer. Methods Partial Differ. Equ. 21(5), 986–997 (2005) · Zbl 1077.76037
[10] Burman, E., Hansbo, P.: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Eng. 195(19–22), 2393–2410 (2006) · Zbl 1125.76038
[11] Burman, E., Hansbo, P.: A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007) · Zbl 1101.76032
[12] Burman, E.: Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Partial Differ. Equ. 24(1), 127–143 (2008) · Zbl 1139.76029
[13] Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Evalu. Eng. 4(4), 308–317 (2001)
[14] Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009) · Zbl 1205.65312
[15] D’Angelo C., Zunino P.: A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47(5), 3990–4020 (2009) · Zbl 1426.76101
[16] Egger H.: A class of hybrid mortar finite element methods for interface problems with non-matching meshes. Technical report, AICES-2009-2 (2009)
[17] Ehrhardt, M., Fuhrmann, J., Linke, A.: A Model of an Electrochemical Flow cell with Porous Layer. Weierstrass Institute for Applied Analysis and Stochastics, Berlin (2009)
[18] Ehrhardt, M., Fuhrmann, J., Holzbecher, E., Linke, A.: Mathematical modeling of channel–porous layer interfaces in PEM fuel cells. In: Conference on Fundamentals and Developments of Fuel Cells, University of Nancy, Nancy, France. ISBN 978-2-7466-0413-1, 10–12th December 2008
[19] Gekeler E.: Mathematische Methoden zur Mechanik. Springer, Berlin (2006) · Zbl 1100.70001
[20] Griebel, M., Klitz, M.: Homogenization and numerical simulation of flow in geometries with textile microstructures. Multiscale Model. Simul. 8(4), 1439–1460 (2010) · Zbl 1383.76445
[21] Hannukainen, A., Juntunen, M., Stenberg, R.: Computations with finite element methods for the Brinkman problem. Comput. Geosci. 15, 155–166 (2011) · Zbl 1333.76051
[22] Hansbo, P., Juntunen, M.: Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. Appl. Numer. Math. 59(6), 1274–1289 (2009) · Zbl 1162.76056
[23] Iliev, O., Lazarov, R., Willems, J.: Discontinuous Galerkin subgrid finite element method for heterogeneous Brinkman’s equations. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) Large-Scale Scientific Computing, vol. 5910 of Lecture Notes in Computer Science, pp. 14–25. Springer, Berlin (2010) · Zbl 1280.76061
[24] Juntunen, M., Stenberg, R.: Analysis of finite element methods for the Brinkman problem. Calcolo 47(3), 129–147 (2010) · Zbl 1410.76179
[25] Kanschat, G., Rivière, B.: A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Comput. Phys. 229(17), 5933–5943 (2010) · Zbl 1425.76068
[26] Kaya, T., Goldak, J.: Three-dimensional numerical analysis of heat and mass transfer in heat pipes. Heat Mass Transf. 43, 775–785 (2007)
[27] Könnö, J., Stenberg, R.: Analysis of H(div)-conforming finite elements for the Brinkman problem. Math. Models Methods Appl. Sci. (2011). doi: 10.1142/S0218202511005726 · Zbl 1331.76115
[28] Könnö, J., Stenberg, R.: 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, pp. 515–522. Springer, Berlin (2010) · Zbl 1311.76062
[29] Lévy, T.: 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)
[30] Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75, 1659–1674 (2006) · Zbl 1119.65110
[31] Mardal, K.A., Tai, X-C., Winther, R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40(5), 1605–1631 (2002) · Zbl 1037.65120
[32] Nédélec, J.-C.: A new family of mixed finite elements in R 3. Numer. Math. 50(1), 57–81 (1986) · Zbl 0625.65107
[33] Nitsche J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971, Collection of articles dedicated to Lothar Collatz on his sixtieth birthday) · Zbl 0229.65079
[34] Popov, P., Efendiev, Y., Qin, G.: Multiscale modeling and simulations of flows in naturally fractured Karst reservoirs. Commun. Comput. Phys. 6(1), 162–184 (2009) · Zbl 1364.76225
[35] Rajagopal K.R.: On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17(2), 215–252 (2007) · Zbl 1123.76066
[36] Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63(1–3), 139–148 (1995) · Zbl 0856.65130
[37] Xie, X., Xu, J., Xue, G.: Uniformly-stable finite element methods for Darcy–Stokes–Brinkman models. J. Comput. Math. 26(3), 437–455 (2008) · Zbl 1174.76013
[38] Xu, X., Zhang, S.: A new divergence-free interpolation operator with applications to the Darcy–Stokes–Brinkman equations. SIAM J. Sci. Comput. 32(2), 855–874 (2010) · Zbl 1352.76071
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.