Research Article

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

  1. Janne Martikainen*,
  2. Tuomo Rossi,
  3. Jari Toivanen

Article first published online: 17 JUL 2001

DOI: 10.1002/cnm.422

Issue

Communications in Numerical Methods in Engineering

Communications in Numerical Methods in Engineering

Volume 17, Issue 8, pages 521–527, August 2001

Additional Information

How to Cite

Martikainen, J., Rossi, T. and Toivanen, J. (2001), Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers. Communications in Numerical Methods in Engineering, 17: 521–527. doi: 10.1002/cnm.422

Author Information

  1. University of Jyväskylä, Department of Mathematical Information Technology, P.O. Box 35 (Agora), FIN-40351 Jyväskylä, Finland

*Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (Agora), FIN-40351 Jyväskylä, Finland

Publication History

  1. Issue published online: 17 JUL 2001
  2. Article first published online: 17 JUL 2001
  3. Manuscript Accepted:
  4. Manuscript Received:

Funded by

  • The Academy of Finland. Grant Numbers: #43066, #66407

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Keywords:

  • algebraic eigenvalue problem;
  • Lanczos algorithm;
  • fast elliptic solver

Abstract

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 differential eigenvalue problems 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 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. Copyright © 2001 John Wiley & Sons, Ltd.

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