Fast Galerkin Method for Solving Helmholtz Boundary Integral Equations on Screens

TL;DR

This paper introduces a fast Galerkin method using spherical harmonic-based discretization for solving Helmholtz boundary integral equations on 3D screens, achieving spectral convergence and demonstrating competitive numerical results.

Contribution

The paper presents a novel Galerkin-Bubnov method with spherical harmonic projections for boundary integral equations, providing spectral convergence and comprehensive error analysis.

Findings

Spectral convergence achieved with the proposed basis functions.

Numerical results comparable to Nyström and hp-methods.

Fully discrete error analysis supports the method's effectiveness.

Abstract

We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the canonical disk. We show that these bases allow for spectral convergence and provide fully discrete error analysis. Numerical experiments support our claims, with results comparable to Nystr\"om-type and $hp$-methods.