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Subject: stabilization parameter × WGL Institute: WIAS ×

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    On the grad-div stabilization for the steady Oseen and Navier-Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ahmed, Naveed
    This paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H1 (Omega) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H1 (Omega) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier-Stokes equations, numerical simulations were performed on a two-dimensional flow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization.
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    SUPG reduced order models for convection-dominated convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Iliescu, Traian; John, Volker; Schyschlowa, Swetlana; Wells, David
    This paper presents a Streamline-Upwind Petrov--Galerkin (SUPG) reduced order model (ROM) based on Proper Orthogonal Decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection-diffusion-reaction equations. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.