Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements

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Date
2015
Volume
2112
Issue
Journal
Series Titel
WIAS Preprints
Book Title
Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and CrankNicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results

Description
Keywords
time-dependent Oseen equations, inf-sup stable pairs of finite element spaces, grad-div stabilization, backward Euler scheme, two-step backward differentiation scheme (BDF2), Crank–Nicolson scheme, uniform error estimates
Citation
Frutos, J. d., García-Archilla, B., John, V., & Novo, J. (2015). Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements (Vol. 2112). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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