Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis
This work addresses high-frequency scattering problems for researchers in numerical analysis and computational physics, but it is incremental as it reviews and applies existing semiclassical techniques rather than introducing new methods.
The paper tackles the numerical solution of high-frequency scattering problems using the Helmholtz equation by applying semiclassical analysis, which provides a phase-space perspective to link solutions to geometric-optic rays, and showcases results on finite-element, boundary-element, and domain-decomposition methods.
We consider the numerical solution of high-frequency scattering problems modeled 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 $\textit{semiclassical analysis}$ 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-optic rays. The goals of the article are to (i) give a introduction to semiclassical analysis aimed at non-experts and (ii) showcase some of the numerical-analysis results about finite-element methods, boundary-element methods, and domain-decomposition methods obtained using semiclassical techniques.