Coordinate complexification for the Helmholtz equation with Dirichlet boundary conditions in a perturbed half-space

TL;DR

This paper introduces a novel complexification method for solving the Helmholtz equation with Dirichlet boundary conditions in perturbed half-spaces, transforming oscillatory integrals into exponentially decaying ones without volumetric modifications.

Contribution

The authors develop a new complexification scheme based on the double layer potential that enables analytic continuation and contour deformation, reducing computational complexity for Helmholtz problems.

Findings

Effective reduction of infinite domain to finite domain via contour deformation

Applicable to 2D and 3D Helmholtz problems with various boundary conditions

Demonstrated improved performance through numerical examples

Abstract

We present a new complexification scheme based on the classical double layer potential for the solution of the Helmholtz equation with Dirichlet boundary conditions in compactly perturbed half-spaces in two and three dimensions. The kernel for the double layer potential is the normal derivative of the free-space Green's function, which has a well-known analytic continuation into the complex plane as a function of both target and source locations. Here, we prove that - when the incident data are analytic and satisfy a precise asymptotic estimate - the solution to the boundary integral equation itself admits an analytic continuation into specific regions of the complex plane, and satisfies a related asymptotic estimate (this class of data includes both plane waves and the field induced by point sources). We then show that, with a carefully chosen contour deformation, the oscillatory integrals are converted to exponentially decaying integrals, effectively reducing the infinite domain to a domain of finite size. Our scheme is different from existing methods that use complex coordinate transformations, such as perfectly matched layers, or absorbing regions, such as the gradual complexification of the governing wavenumber. More precisely, in our method, we are still solving a boundary integral equation, albeit on a truncated, complexified version of the original boundary. In other words, no volumetric/domain modifications are introduced. The scheme can be extended to other boundary conditions, to open wave guides and to layered media. We illustrate the performance of the scheme with two and three dimensional examples.