Konferenzraum 614 (Mathegebäude)

We consider linear systems arising from the linear stability analysis of solu- tions of the Navier-Stokes equations. Such stability analysis leads to the solution of an eigenvalue problem, in particular, to the determination of eigenvalues close to the imaginary axis. Shift-and-invert type methods are often used for the so- lution of the eigenvalue problem, leading (on the continuous level) to systems of the form: Given a mean velocity field U, a forcing term f , a scalar  > 0 and a viscosity coefficient , find a velocity-pressure pair u, p which solves −
u − u + (U · ∇)u + (u · ∇)U + ∇p = f in divu = 0 in u = 0 on @ Due to indefiniteness of the submatrix corresponding to the velocities, this sys- tem poses a serious challenge for iterative solution methods. In this talk we discuss the extension of the augmented Lagrangian-based block triangular pre- conditioner to this class of problems. We prove eigenvalue estimates for the velocity submatrix and deduce several representations of the Schur complement operator which are relevant to numerical properties of the augmented system. Numerical experiments on several model problems demonstrate the effectiveness and robustness of the preconditioner over a wide range of problem parameters. This is part of a joint research with M.Benzi from Emory.

Prof. Maxim A. Olshanskii

Moscow State University