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A two-level pressure stabilization method for the generalized Stokes problem. (English) Zbl 1331.76072

Summary: Stabilization mixed methods that circumvent the restrictive inf-sup condition without introducing penalty errors have been developed for Stokes equations e.g. by L. P. Franca, T. J. R. Hughes and R. Stenberg [in: Incompressible computational fluid dynamics: trends and advances. Cambridge: Cambridge University Press, 87–107 (1993; Zbl 1189.76339)], and J. Bonvin, M. Picasso and R. Stenberg [Comput. Methods Appl. Mech. Eng. 190, No. 29–30, 3893–3914 (2001; Zbl 1014.76043)]. These methods consist of modifying the standard Galerkin formulation by adding mesh-dependent terms, which are weighted residuals of the original differential equations. The aim of the stabilization, however, is to select minimal terms that stabilize the approximation without losing the nice conservation properties.
Although for properly chosen stabilization parameters these methods are well posed for all velocity-pressure pairs, numerical results reported by several researchers seem to indicate that these methods are sensitive to the choice of the stabilization parameters. A relatively recent stabilized finite-element formulation that seems less sensitive to the choice of parameters and has better local conservation properties was developed and analysed by R. Codina and J. Blasco [Comput. Methods Appl. Mech. Eng. 143, No. 3–4, 373–391 (1997; Zbl 0893.76040)], R. Becker and M. Braack [Calcolo 38, No. 4, 173–199 (2001; Zbl 1008.76036)], and K. Nafa [in: ICNAAM 2004. International conference on numerical analysis and applied mathematics 2004, Chalkis, Greece. Weinheim: Wiley-VCH, 280–282 (2004; Zbl 1187.76690)]. This method consists of introducing the \(L^2\)-projection of the pressure gradient as a new unknown of the problem. Hence, a third equation to enforce the projection property is added to the original discrete equations, and a weighted difference of the pressure gradient and its projection are introduced into the continuity equation. In this paper, as done by K. Nafa [Int. J. Numer. Methods Fluids 56, No. 6, 753–765 (2008; Zbl 1130.76048)], we analyse the pressure gradient stabilization method for the generalized Stokes problem and investigate its stability and convergence properties.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

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[3] DOI: 10.1016/0045-7825(86)90025-3 · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
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