×

\(H(\mathop{div})\)-conforming finite elements for the Brinkman problem. (English) Zbl 1331.76115

Summary: The Brinkman equations describe the flow of a viscous fluid in a porous matrix. Mathematically the Brinkman model is a parameter-dependent combination of both the Darcy and Stokes models. We introduce a dual mixed framework for the problem, and use \(H(\mathop {div})\)-conforming finite elements with the symmetric interior penalty Galerkin method to obtain a stable formulation. We show that the formulation is stable in a mesh-dependent norm for all values of the parameter. We also introduce a postprocessing scheme for the pressure along with a residual-based a posteriori estimator, which is shown to be efficient and reliable for all parameter values.

MSC:

76S05 Flows in porous media; filtration; seepage
76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF00375065 · Zbl 0724.76020 · doi:10.1007/BF00375065
[2] DOI: 10.1007/BF00375066 · Zbl 0724.76021 · doi:10.1007/BF00375066
[3] DOI: 10.1007/s10596-006-9024-8 · Zbl 1197.76122 · doi:10.1007/s10596-006-9024-8
[4] DOI: 10.1137/0719052 · Zbl 0482.65060 · doi:10.1137/0719052
[5] Arnold D. N., RAIRO Modél. Math. Anal. Numér. 19 pp 7–
[6] DOI: 10.1137/08072632X · Zbl 1406.76047 · doi:10.1137/08072632X
[7] DOI: 10.1137/S0036142994264079 · Zbl 0866.65071 · doi:10.1137/S0036142994264079
[8] DOI: 10.1007/BF01396752 · Zbl 0631.65107 · doi:10.1007/BF01396752
[9] DOI: 10.1007/978-1-4612-3172-1 · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[10] DOI: 10.1007/s10915-006-9107-7 · Zbl 1151.76527 · doi:10.1007/s10915-006-9107-7
[11] DOI: 10.1137/080726318 · Zbl 1426.76101 · doi:10.1137/080726318
[12] DOI: 10.1007/s10596-010-9204-4 · Zbl 1333.76051 · doi:10.1007/s10596-010-9204-4
[13] Houston P., J. Sci. Comput. 23 pp 347–
[14] DOI: 10.1007/978-3-642-12535-5_2 · Zbl 1280.76061 · doi:10.1007/978-3-642-12535-5_2
[15] DOI: 10.1007/s10092-009-0017-6 · Zbl 1410.76179 · doi:10.1007/s10092-009-0017-6
[16] DOI: 10.1016/j.jcp.2010.04.021 · Zbl 1425.76068 · doi:10.1016/j.jcp.2010.04.021
[17] DOI: 10.1002/fld.1795 · Zbl 1140.76020 · doi:10.1002/fld.1795
[18] DOI: 10.1007/s00231-006-0166-y · doi:10.1007/s00231-006-0166-y
[19] DOI: 10.1090/S0025-5718-06-01872-2 · Zbl 1119.65110 · doi:10.1090/S0025-5718-06-01872-2
[20] DOI: 10.1007/BF01389668 · Zbl 0625.65107 · doi:10.1007/BF01389668
[21] DOI: 10.1007/BF02995904 · Zbl 0229.65079 · doi:10.1007/BF02995904
[22] DOI: 10.1142/S0218202507001899 · Zbl 1123.76066 · doi:10.1142/S0218202507001899
[23] DOI: 10.1090/S0025-5718-96-00720-X · Zbl 0857.65117 · doi:10.1090/S0025-5718-96-00720-X
[24] DOI: 10.1007/BF01389651 · Zbl 0563.65072 · doi:10.1007/BF01389651
[25] DOI: 10.1007/BF01397550 · Zbl 0632.73063 · doi:10.1007/BF01397550
[26] DOI: 10.1007/BF01405291 · Zbl 0708.76088 · doi:10.1007/BF01405291
[27] Stenberg R., RAIRO Modél. Math. Anal. Numér. 25 pp 151– · Zbl 0717.65081 · doi:10.1051/m2an/1991250101511
[28] DOI: 10.1016/0377-0427(95)00057-7 · Zbl 0856.65130 · doi:10.1016/0377-0427(95)00057-7
[29] DOI: 10.1007/s00211-009-0272-0 · Zbl 1189.65285 · doi:10.1007/s00211-009-0272-0
[30] Verfürth R., A Review of a posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996) · Zbl 0853.65108
[31] DOI: 10.1137/080730044 · Zbl 1407.76074 · doi:10.1137/080730044
[32] DOI: 10.1137/060649227 · Zbl 1138.76049 · doi:10.1137/060649227
[33] Xie X., J. Comput. Math. 26 pp 437–
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.