Stabilized finite element schemes with LBB-stable elements for incompressible flows

Authors:
Tobias Gelhard;Gert Lube;Maxim A. Olshanskii;Jan-Hendrik Starcke
Affiliations:
Mathematics Department, University of Göttingen, D-37083, Germany;Mathematics Department, University of Göttingen, D-37083, Germany;Department of Mechanics and Mathematics, Moscow M.V. Lomonosov University, Moscow 119899, Russia;Mathematics Department, University of Göttingen, D-37083, Germany
Venue:
Journal of Computational and Applied Mathematics
Year:
2005

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Abstract

We study stabilized FE approximations of SUPG type to the incompressible Navier-Stokes problem. Revisiting the analysis for the linearized model, we show that for conforming LBB-stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal-order approximation. Then we analyze a reduced SUPG scheme often used in practice for LBB-stable elements. To provide the reduced scheme with appropriate stability estimates we introduce a modified LBB condition which is proved for a family of FE approximations. The analysis is given for the linearized equations. Numerical experiments for some linear and nonlinear benchmark problems support the theoretical results.