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

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 $\text{semiclassical analysis}$ 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.

Review article, 173 pages

References in corpus (11)

• A rapidly converging domain decomposition method for the Helmholtz equation
• Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation
• A Survey of Trefftz Methods for the Helmholtz Equation
• On stability of discretizations of the Helmholtz equation (extended version)
• Strong unique continuation for general elliptic equations in 2D
• On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients
• Optimal constants in nontrapping resolvent estimates and applications in numerical analysis
• Schwarz methods by domain truncation
• An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation
• Wavenumber-explicit $hp$-FEM analysis for Maxwell's equations with transparent boundary conditions
• A Unified Framework for Double Sweep Methods for the Helmholtz Equation