The Jacobi-Davidson method is suitable for computing solutions of high dimensional eigenvalue problems. It is an iterative method that improves the approximate eigenvectors and eigenvalues by exploiting approximate solutions of specific linear systems, the so-called correction equation. This equation allows preconditioning, which makes Jacobi-Davidson a powerful method.
We discuss approaches for an efficient handling of the correction equation.
For PDE related problems, we propose a solution strategy based on a non-overlapping domain-decomposition technique. For a model eigenvalue problem we derive optimal coupling parameters. By numerical examples we outline how to apply the developed theory to more general problems and show the effect of this approach on the overall Jacobi-Davidson process.
In case of excellent preconditioners, the obvious approach may introduce instabilities that may hamper convergence. For preconditioners of multilevel type, we show how such instabilities can be avoided. We explain also that such preconditioners can efficiently and effectively be updated when better approximations to the eigenvalues of interest become available. The approach also leads to good initial guesses for the eigenvectors.