New Insights on High Wave Scattering by Multiple Open Arcs: Exponentially Convergent Methods and Shape Holomorphy

In this talk, we focus on the scattering of time-harmonic acoustic, elastic, and polarized electromagnetic waves by multiple finite-length open arcs in an unbounded two-dimensional domain. We begin by reformulating the corresponding boundary value problems with Dirichlet or Neumann conditions as weakly and hypersingular boundary integral equations (BIEs), respectively. We then introduce a family of fast spectral Galerkin methods for solving these BIEs. The discretization bases are built from weighted Chebyshev polynomials that accurately capture the solutions’ edge behavior. Under the assumption of analyticity of the sources and arc geometries, we show that these bases yield exponential convergence with respect to the polynomial degree.

Numerical examples will illustrate the accuracy and robustness of the proposed methods, with respect to both the number of arcs and the wavenumber. Additionally, we demonstrate that, for general weakly and hypersingular BIEs, the solutions depend holomorphically on perturbations of the arc parametrizations. These results are crucial for establishing the shape holomorphy of domain-to-solution maps arising in boundary integral equations, with applications in uncertainty quantification, inverse problems, and deep learning, among others. They also raise new questions—some of which you may have the answer to!

• Speaker: Carlos Jerez Hanckes (University of Bath)
• Thursday 15 May 2025, 15:00–16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.