An Augmented Lagrangian Approach to Linearized Problems in Hydrodynamic Stability

The solution of linear systems arising from the linear stability analysis of solutions of the Navier-Stokes equations is considered. Due to indefiniteness of the submatrix corresponding to the velocities, these systems pose a serious challenge for iterative solution methods. In this paper, the augmented Lagrangian-based block triangular preconditioner introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113] is extended 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.

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