A Class of Spectral Two-Level Preconditioners

When solving the linear system Ax = b with a Krylov method, the smallest eigenvalues of the matrix A often slow down the convergence. This is usually still the case even after the system has been preconditioned. Consequently if the smallest eigenvalues of A could be somehow “removed” the convergence would be improved. Several techniques have been proposed in the past few years that attempt to tackle this problem. The proposed approaches can be split into two main families depending on whether the scheme enlarges the generated Krylov space or adaptively updates the preconditioner. In this paper, we follow the second approach and propose a class of preconditioners both for unsymmetric and for symmetric linear systems that can also be adapted for symmetric positive definite problems. Our preconditioners are particularly suitable when there are only a few eigenvalues near the origin that are well separated. We show that our preconditioners shift these eigenvalues from close to the origin to near one. We illustrate the performance of our method through extensive numerical experiments on a set of general linear systems. Finally we show the advantages of the preconditioners for solving dense linear systems arising in electromagnetism applications that were the main motivation for this work.