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Author: John, Volker × Has files: Yes × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: backward Euler scheme × WGL Institute: WIAS ×

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    Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Frutos, Javier de; García-Archilla, Bosco; John, Volker; Novo, Julia
    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
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    Error analysis of the SUPG finite element disretization of evolutionary convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) John, Volker; Novo, Julia
    Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the SUPG finite element stabilization of time-dependent convection-diffusion-reaction equations that is combined with the backward Euler method. Standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experiences, this seems not to be the correct behavior. For this reason, the time-continuous case is analyzed under certain conditions on the coefficients of the equation and the finite element method. An error estimate with the standard order of convergence is derived for stabilization parameters of the same form that is optimal for the steady-state problem. Numerical studies support the analytical results.