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An Arnoldi-based iterative scheme for nonsymmetric matrix pencils arising in finite element stability problems. (English) Zbl 0759.65070
There are three algorithms which are used for the computation of several eigenpairs: a) the subspace iteration, b) the unsymmetric Lanczos procedure, and c) the Arnoldi iteration. The latter which is described in an unpublished paper by Y. Saad, is the favorite candidate in this paper. An implementation with appropriate restarts and shifts is presented. The application to the following two real life problems is discussed: the Orr-Sommerfeld equation and a chemical reactor equation.
The discussion contains the following remark: ‘The finite element discretization of continuous eigenvalue problems often leads to the appearance of a large number of “spurious” eigenmodes in the discrete problem with corresponding eigenvalues having very large modulus. These spurious eigenmodes have no counterpart in the original continuous problem and are therefore extremely sensitive and ill-conditioned with respect to the discretization’.
Reviewer: D.Braess (Bochum)

MSC:
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
35P15 Estimates of eigenvalues in context of PDEs
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