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Continuous interior penalty finite element method for Oseen’s equations. (English) Zbl 1344.76049

Summary: We present an extension of the continuous interior penalty method of J. Douglas jun. and T. Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods. Lect. Notes Phys. 58, 207–216 (1976; doi:10.1007/BFb0120591)] to Oseen’s equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving \(L^2\)-control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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