Boundary value problems for elliptic partial differential operators can be solved very efficiently by using multigrid solvers or multigrid preconditioned iterations. In the past 25 years such iterations have gained an enormous importance and deep theoretical results have been derived. In contrast to this, the analysis for eigenvalue problems for the same elliptic operators has not reached an equivalent state of development. The paper deals with just these eigenvalue problems for self-adjoint and coercive elliptic partial differential operators. Most of the interest for this paper may be explained as follows: the present paper shows how to use well-known methods for solving boundary value problems to solve the corresponding eigenvalue problems. The new analysis has made clear that essentially the same convergence behavior can be expected for the eigenvalue iteration. That means, e.g., that grid-independent convergence rates for multigrid preconditioned eigensolvers are possible. This direct link between solvers for the boundary value problem and solvers for the eigenvalue problem is a crucial point of the paper. Around it, sharp convergence estimates have been derived by taking advantage of a geometric interpretation of the iterative eigensolver.

A practically and theoretically interesting aspect of the paper is that it reveals a close relation between solvers for the boundary value problems and those for the eigenvalue problems for self-adjoint and coercive elliptic partial differential operators. On the one hand, existing program code for solving boundary value problems (multigrid iterations and multigrid preconditioners) can easily be used to solve the corresponding eigenvalue problems. On the other hand, the convergence analysis of preconditioned eigensolvers is mainly built on a new geometric interpretation. Moreover, an error propagation equation for an iterative eigensolver has been suggested. This approach may open new ways of understanding and analyzing comparable and improved iterative solvers for eigenvalue problems.

Eigenvalue problems appear in several applications, both theoretical and practical. For instance, the vibrations of a turbine or the aero-elastic vibrations of a bridge can be described by eigenvalues (i.e. the frequencies of vibration) and eigenvectors (i.e. modes of vibration). The precise computation of such eigenvalues/vectors for large mechanical structures requires immense computer power and specially designed algorithms. The paper analyzes an iterative method for these eigenvalue problems and shows that efficient techniques, which are well known for linear systems of equations (boundary value problems), can be used in a very similar way to solve eigenvalue problems.

The paper is the first part of a series of three papers containing results of my habilitation thesis which I finished in Tuebingen in September 2001. Initially, I started the work on eigenvalue problems in a project of the Sonderforschungsbereich 382 (Collaborative Research Centre of the Deutsche Forschungsgemeinschaft).