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Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. (English) Zbl 1125.76038

Summary: We introduce and analyze a stabilized finite element method for the generalized Stokes equation. Stability is obtained by adding a least squares penalization of the gradient jumps across element boundaries. The method can be seen as a higher order version of the Brezzi-Pitkäranta penalty stabilization [F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in: W. Hackbusch (ed.), Efficient solution of elliptic systems, Vieweg (1984)], but gives better resolution on the boundary for Stokes equation than does classical Galerkin least-squares formulation. We prove optimal and quasi-optimal convergence properties for Stokes problem and for porous media models of Darcy and Brinkman. Some numerical examples are given.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76S05 Flows in porous media; filtration; seepage
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