Field of Values Analysis of a Two-Level Preconditioner for the Helmholtz Equation

In this paper, we study the convergence of a two-level preconditioned GMRES for linear systems related to first order finite element discretizations of Helmholtz equation in a lossy media. Due to losses, the finite element system matrix is nonnormal. To handle this nonnormality, we use a field of values based convergence criterion for GMRES. The focus is on a priori analysis to study the dependency between GMRES convergence and wave number, losses, and coarse as well as fine grid mesh sizes, before any actual computations are done. The analysis indicates that the coarse grid mesh size $H$ should satisfy the constraint $\kappa^3 H \ll 1$ to guarantee wave number and mesh size independent convergence of the preconditioned iteration. The obtained theoretical results are illustrated in two- and three-dimensional numerical examples.

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