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block matrices; eigenvalues; Cayley transform; Navier-Stokes; large sparse generalized eigenvalue problems; Hopf bifurcations
This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.
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