Applying GMRES to Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

We are interested in solving scattering problems with strong trapping using the Combined Field Integral Equation (CFIE) and the Generalized Minimal Residual method (GMRes). In this talk, we show a new understanding of how the number of GMRes iterations depends on frequency in this situation. The non-normal nature of CFIE makes GMRes the iterative method of choice for solving linear systems stemming from its discretisation. GMRes has the advantage of being able to solve any non-singular linear system, in particular non-normal. But the convergence analysis becomes less straightforward in this case, because it requires more information than just the spectrum. Bounds for GMRes applied to non-normal matrices can be derived using condition number of eigenvalues, numerical range or pseudo-spectrum. But in the case of trapping, an additional difficulty comes from the solution operator whose norm grows exponentially through a sequence of frequencies tending to infinity, with the density of these ``bad'' frequencies increasing as the frequency increases. In this case, the spectrum of the associated matrix has the form of a cluster associated with eigenvalues near the origin. We provide a new analysis of the GMRes convergence taking into account this particular distribution, which allows to show more precisely why the number of iterations grows with the frequency. This is joint work with A. Spence et E. A. Spence.