Abstract
This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldi's method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.
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Cliffe, K.A., Garratt, T.J. & Spence, A. Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations. Adv Comput Math 1, 337–356 (1993). https://doi.org/10.1007/BF02072015
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DOI: https://doi.org/10.1007/BF02072015