Scattering Problems for Engineering

Wave propagation problems on unbounded domains are ubiquitous in computational mathematics, engineering, and technology, appearing in various domains such as ultrasonics, seismics, underwater acoustics, and electrodynamics. As a concrete example, consider the numerical computation of the radar cross-section of aircraft. Maxwell equations are solved numerically for the interaction of an incident electromagnetic wave with the aircraft.

Scattering observables depend on the radiation field far from the source. My approach uses compactification and a rescaling of the outgoing field to make infinity a computational boundary, giving direct access to far-field information.

Results and current work

A null infinity layer for wave scattering (SIAM Journal on Scientific Computing, 2026) develops this approach for time-harmonic scattering. The transformations are restricted to a layer around the interior domain, and the paper demonstrates several numerical discretizations in one and two dimensions.

With Markus Wess, I extend the construction to conforming finite elements in Finite Elements for Helmholtz Scattering with Infinity as a Computational Boundary (2026 preprint). We compare the method with perfectly matched layers in two- and three-dimensional Helmholtz benchmarks, including trapping geometries and a submarine benchmark.

From Penrose to Melrose: Computing Scattering Amplitudes at Infinity for Unbounded Media (2026 preprint) treats variable media, including media with long-range asymptotics. It constructs a solver near infinity that can be coupled to an interior solver through domain decomposition.

Online book and code

Our book Hyperboloidal Compactification in NGSolve explains the finite element implementation and includes executable frequency-domain and time-domain examples. The hypFEM repository contains the source code. The demonstrated Helmholtz problems provide a basis for exploring further applications in acoustics and other areas of wave propagation.