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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.

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
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

Citations:

Zbl 1331.76115
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References:

[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
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