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arXiv · 2511.15287

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

Jeffrey Galkowski·Euan A. Spence
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Abstract

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.

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Jeffrey Galkowski, Euan A. Spence. 2025-11-19. Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis. https://doi.org/10.1017/s0962492925100275

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