# Numerical Analysis and Scientific Computing Seminar

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

**Speaker:**
Euan Spence, University of Bath

**Location:**
Warren Weaver Hall online

**Date:**
April 23, 2021, 10 a.m.

**Synopsis:**

This talk will consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping (e.g. in the exterior of a "horse-shoe-shaped" domain which can trap rays).

This problem is interesting/difficult because it is exponentially ill-conditioned through an increasing sequence of frequencies.

I will present upper bounds on how the number of GMRES iterations grows with the frequency, proved under certain assumptions about the distribution of the eigenvalues (and I will discuss to what extent these assumptions are proved).

This talk will be primarily concerned with boundary-integral-equation formulations of the Helmholtz equation, but no a priori knowledge of boundary-integral equations will be assumed.

This is joint work with Pierre Marchand (Bath), Jeff Galkowski (UCL), and Alastair Spence (Bath).