Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

摘要

We consider the Jacobi preconditioner of the GMRES method introduced by Liu and Jin for the scattering problem (IEEE Trans. Ante. Prop. 2002; 50:132-140). We explain why it is a particular form of the Schwarz' preconditioner with a complete overlap and specific transmission conditions. So far, a superlinear convergence has been predicted by the general theory without any additional indication on the convergence rates. Here, we establish error bounds that provide accurate convergence rates in two and three dimensions. Courant-Weyl's min-max principle applied to some kernel operators together with some polynomial approximation estimates are the milestones for the proofs.