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A computational study of stabilized, low-order \(C^{0}\) finite element approximations of Darcy equations. (English) Zbl 1177.76191
Summary: We consider finite element methods for Darcy equations that are designed to work with standard, low order \(C ^{0}\) finite element spaces. Such spaces remain a popular choice in engineering practice because they offer the convenience of simple and uniform data structures and reasonable accuracy. A consistently stabilized method [A. Masud and T. J. R. Hughes, Comput. Methods Appl. Mech. Eng. 191, No. 39–40, 4341–4370 (2002; Zbl 1015.76047)] and a least-squares formulation [D. C. Jespersen, Math. Comput. 31, 873–880 (1977; Zbl 0383.65060)] are compared with two new stabilized methods. The first one is an extension of a recently proposed polynomial pressure projection stabilization of the Stokes equations [the authors and M. D. Gunzburger, SIAM J. Numer. Anal. 44, No. 1, 82–101 (2006; Zbl 1145.76015); the authors, Int. J. Numer. Methods Fluids 46, No. 2, 183–201 (2004; Zbl 1060.76569)]. The second one is a weighted average of a mixed and a Galerkin principles for the Darcy problem, and can be viewed as a consistent version of the classical penalty stabilization for the Stokes equations [F. Brezzi et al., J. Sci. Comput. 22–23, 119–145 (2005; Zbl 1103.76031)]. Our main conclusion is that polynomial pressure projection stabilization is a viable stabilization choice for low order \(C ^{0}\) approximations of the Darcy problem.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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