Continuous Interior Penalty Finite Element Method for Oseen's Equations

Abstract : In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207–216] 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 pres-sure/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.
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Erik Burman, Miguel Angel Fernández, Peter Hansbo. Continuous Interior Penalty Finite Element Method for Oseen's Equations. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 44 (3), pp.1248-1274. ⟨10.1137/040617686⟩. ⟨hal-01345519⟩

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