A Dual–Mixed Finite Element Method for the Brinkman Problem
The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 1-17.

A mixed variational formulation of the Brinkman problem is presented which is uniformly well–posed for degenerate (vanishing) coefficients under the hypothesis that a generalized Poincaré inequality holds. The construction of finite element schemes which inherit this property is then considered.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.7
Classification: 65N30,  65N12
Keywords: Brinkman, Stokes, Darcy, mixed methods
@article{SMAI-JCM_2016__2__1_0,
     author = {Jason S. Howell and Michael Neilan and Noel J. Walkington},
     title = {A {Dual{\textendash}Mixed} {Finite} {Element} {Method} for the {Brinkman} {Problem}},
     journal = {The SMAI journal of computational mathematics},
     pages = {1--17},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {2},
     year = {2016},
     doi = {10.5802/smai-jcm.7},
     mrnumber = {3633543},
     zbl = {1416.76112},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.7/}
}
Jason S. Howell; Michael Neilan; Noel J. Walkington. A Dual–Mixed Finite Element Method for the Brinkman Problem. The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 1-17. doi : 10.5802/smai-jcm.7. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.7/

[1] G. Allaire Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., Volume 113 (1990) no. 3, pp. 209-259 | Article | MR 1079189

[2] G. Allaire 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. Rational Mech. Anal., Volume 113 (1990) no. 3, pp. 261-298 | Article | MR 1079190

[3] F. Anisi; N. Salehi-Nik; G. Amoabediny; B. Pouran; N. Haghighipour; B. Zandieh-Doulabi Applying shear stress to endothelial cells in a new perfusion chamber: hydrodynamic analysis, Journal of Artificial Organs, Volume 17 (2014) no. 4, pp. 329-336 | Article

[4] T. Arbogast; H.L. Lehr Homogenization of a Darcy-Stokes system modeling vuggy porous media, Comput. Geosci., Volume 10 (2006) no. 3, pp. 291-302 | Article | MR 2261837 | Zbl 1197.76122

[5] D.N. Arnold; J. Jr. Douglas; C.P. Gupta A family of higher order mixed finite element methods for plane elasticity, Numer. Math., Volume 45 (1984) no. 1, pp. 1-22 | Article | MR 761879

[6] D. Boffi; F. Brezzi; L.F. Demkowicz; R.G. Durán; R.S. Falk; M. Fortin Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, Volume 1939, Springer-Verlag, Berlin, 2008, x+235 pages (Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006, Edited by Boffi and Lucia Gastaldi) | Article | MR 2459075

[7] F. Brezzi; J. Jr. Douglas; M. Fortin; L.D. Marini Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér., Volume 21 (1987) no. 4, pp. 581-604 | Article | Numdam | MR 921828

[8] F. Brezzi; M. Fortin Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Volume 15, Springer-Verlag, New York, 1991, x+350 pages | MR 1115205 | Zbl 0788.73002

[9] N. Chen; M. Gunzburger; W. Wang Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system, Journal of Mathematical Analysis and Applications, Volume 368 (2010) no. 2, pp. 658 -676 http://www.sciencedirect.com/science/article/pii/S0022247X10001472 | Article | MR 2643831 | Zbl 1352.35093

[10] D.B. Das Hydrodynamic modelling for groundwater flow through permeable reactive barriers, Hydrological Processes, Volume 16 (2002) no. 17, pp. 3393-3418 | Article

[11] G. N. Gatica; L.F. Gatica; A. Márquez Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow, Numer. Math., Volume 126 (2014) no. 4, pp. 635-677 | Article | MR 3175180 | Zbl 1426.74232

[12] G. N. Gatica; R. Oyarzúa; F.-J. Sayas Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem, Numer. Methods Partial Differential Equations, Volume 27 (2011) no. 3, pp. 721-748 | Article | MR 2809968 | Zbl 1301.76046

[13] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984, xv+493 pages | Article

[14] M. Griebel; M. Klitz Homogenization and numerical simulation of flow in geometries with textile microstructures, Multiscale Model. Simul., Volume 8 (2010) no. 4, pp. 1439-1460 | Article | MR 2718267 | Zbl 1383.76445

[15] J. Guzmán; M. Neilan A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal., Volume 32 (2012) no. 4, pp. 1484-1508 | Article | MR 2991835 | Zbl 1332.76057

[16] A. Hannukainen; M. Juntunen; R. Stenberg Computations with finite element methods for the Brinkman problem, Comput. Geosci., Volume 15 (2011) no. 1, pp. 155-166 | Article | Zbl 1333.76051

[17] J.S. Howell; N.J. Walkington Inf-sup conditions for twofold saddle point problems, Numer. Math., Volume 118 (2011) no. 4, pp. 663-693 | Article | MR 2822495 | Zbl 1230.65128

[18] J.S. Howell; N.J. Walkington Dual-mixed finite element methods for the Navier-Stokes equations, ESAIM: M2AN, Volume 47 (2013) no. 3, pp. 789-805 | Article | Numdam | MR 3056409

[19] M. Juntunen; R. Stenberg Analysis of finite element methods for the Brinkman problem, Calcolo, Volume 47 (2010) no. 3, pp. 129-147 | Article | MR 2672618 | Zbl 1410.76179

[20] G. Kanschat; B. Rivière A strongly conservative finite element method for the coupling of Stokes and Darcy flow, J. Comput. Phys., Volume 229 (2010) no. 17, pp. 5933-5943 | Article | MR 2657851 | Zbl 1425.76068

[21] T. Kaya; J. Goldak Three-dimensional numerical analysis of heat and mass transfer in heat pipes, Heat and Mass Transfer, Volume 43 (2007) no. 8, pp. 775-785 | Article

[22] J. Könnö; R. Stenberg H( div )-conforming finite elements for the Brinkman problem, Math. Models Methods Appl. Sci., Volume 21 (2011) no. 11, pp. 2227-2248 | Article | MR 2860674 | Zbl 1331.76115

[23] J. Könnö; R. Stenberg Numerical computations with H( div )-finite elements for the Brinkman problem, Comput. Geosci., Volume 16 (2012) no. 1, pp. 139-158 | Article | MR 2868741

[24] T. Lévy Erratum: “Loi de Darcy ou loi de Brinkman?”, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, Volume 292 (1981) no. 17, 1239 pages | MR 623960

[25] K. Mardal; X. Tai; R. Winther A Robust Finite Element Method for Darcy-Stokes Flow, SIAM Journal on Numerical Analysis, Volume 40 (2002) no. 5, pp. 1605-1631 | Article | MR 1950614 | Zbl 1037.65120

[26] A. Márquez; S. Meddahi; F.-J. Sayas Strong coupling of finite element methods for the Stokes-Darcy problem, IMA J. Numer. Anal., Volume 35 (2015) no. 2, pp. 969-988 | Article | MR 3335232 | Zbl 1312.76030

[27] P. Monk Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003, xiv+450 pages | Article | Zbl 1024.78009

[28] V. Nassehi Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration, Chemical Engineering Science, Volume 53 (1998) no. 6, pp. 1253-1265 | Article

[29] J.-C. Nédélec Mixed finite elements in R 3 , Numer. Math., Volume 35 (1980) no. 3, pp. 315-341 | Article

[30] J. T. Podichetty; P.R. Bhaskar; A. Khalf; S.V. Madihally Modeling pressure drop using generalized scaffold characteristics in an axial-flow bioreactor for soft tissue regeneration, Annals of Biomedical Engineering, Volume 42 (2014) no. 6, pp. 1319-1330 | Article

[31] R. Qi; X. Yang A nonconforming rectangular finite element pair for the Darcy-Stokes-Brinkman model, Numerical Methods for Partial Differential Equations, Volume 29 (2013) no. 2, pp. 510-530 | Article | MR 3022897 | Zbl 1364.76095

[32] K.R. Rajagopal On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Methods Appl. Sci., Volume 17 (2007) no. 2, pp. 215-252 | Article | MR 2292356

[33] P.-A. Raviart; J.M. Thomas A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Springer, Berlin, 1977, p. 292-315. Lecture Notes in Math., Vol. 606 | Zbl 0362.65089

[34] P. Seokwon; V.D. Dhananjay; V.M. Sundararajan Computational simulation modelling of bioreactor configurations for regenerating human bladder, Computer Methods in Biomechanics and Biomedical Engineering, Volume 16 (2013) no. 8, pp. 840-851 (PMID: 22224865) | Article

[35] X.-C. Tai; R. Winther A discrete de Rham complex with enhanced smoothness, Calcolo, Volume 43 (2006) no. 4, pp. 287-306 | Article | MR 2283095 | Zbl 1168.76311

[36] P. Vassilevski; U. Villa A Mixed Formulation for the Brinkman Problem, SIAM Journal on Numerical Analysis, Volume 52 (2014) no. 1, pp. 258-281 | Article | MR 3162407 | Zbl 06296640

[37] X. Xie; J. Xu; G. Xue Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math., Volume 26 (2008) no. 3, pp. 437-455 | MR 2421892 | Zbl 1174.76013

[38] X. Xu; S. Zhang A new divergence-free interpolation operator with applications to the Darcy-Stokes-Brinkman equations, SIAM J. Sci. Comput., Volume 32 (2010) no. 2, pp. 855-874 | Article | MR 2609343 | Zbl 1352.76071