Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis

We consider the numerical solution of high-frequency scattering problems modelled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and techniques from semiclassical analysis have provided a new perspective and been used to settle several long-standing open problems in this area. Semiclassical analysis works in phase space (i.e. position and frequency) and describes rigorously the extent to which solutions of high-frequency PDEs are dictated by the properties of the corresponding geometric optics rays.

The goals of the article are (i) to give a introduction to semiclassical analysis aimed at non-experts and (ii) to showcase some of the numerical analysis results about finite element methods, boundary element methods and domain decomposition methods obtained using semiclassical techniques.