COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun

SUMMARY The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial diierential eigenvalue problems in two-dimensional or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos algorithm that the original problem. In these experiments, the underlying Poisson and elasticity problems are solved using a standard multigrid method.