Computation of a few smallest eigenvalues of elliptic operators using fastelliptic solvers

The computation of a few smallest eigenvalues of generalized algebraic eige nvalue problems is studied. The considered problems are obtained by discret izing self-adjoint second-order elliptic partial differential eigenvalue pr oblems in two- 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 ari sing linear problems are solved using some standard fast elliptic solver. N umerical experiments demonstrate that the inverted problem is much easier t o solve with the Lanczos algorithm that the original problem. In these expe riments, the underlying Poisson and elasticity problems are solved using a standard multigrid method. Copyright (C) 2001 John Wiley & Sons, Ltd.