CONVERGENCE BOUNDS FOR PRECONDITIONED GMRES USING ELEMENT-BY-ELEMENT ESTIMATES OF THE FIELD OF VALUES

By combining element-by-element estimates for the field of values of a preconditioned matrix with GMRES-convergence estimates it is possible to derive an easily computable upper bound on the GMRES-residual norm. This method can be applied to general finite element systems, but the preconditioner has to be Hermitian and positive definite. The resulting upper bound for the GMRES-residual norm can be used to analyse a given preconditioner, or to optimize a parameter dependent preconditioner. In this paper we will apply this approach to derive a suitable shift for a so-called shifted Laplace preconditioner for the damped Helmholtz equation. Numerical experiments show that the shift that is derived in this way is close to optimal.