Spectral Galerkin Method for Solving Helmholtz and Laplace Dirichlet Problems on Multiple Open Arcs

Abstract

AbstractWe present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz domain $$\mathbb {R}^2 \backslash \overline {\Gamma }$$ ℝ 2 ∖ Γ ¯ , where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is discretized using weighted Chebyshev polynomials. This choice of basis allows for exponential convergence rates under smoothness assumptions. Moreover, by implementing a simple compression algorithm, we are able to efficiently account for large numbers of arcs as well as a wide wavenumber range.