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

TL;DR

This work introduces a coordinate complexification scheme for solving Helmholtz Dirichlet problems in perturbed half-spaces by recasting the solution with a boundary double-layer potential and proving the analytic extension of the density to complex contours. A contour-deformed boundary integral equation is shown to be well-posed, with the complexified operator compact and invertible; truncated complex contours yield exponentially small errors, enabling high-accuracy solutions with logarithmic-in-$1/$ degrees of freedom. The authors provide rigorous proofs, extend the framework to three dimensions, and validate the method through extensive numerical examples, including 2D accuracy tests, complex interfaces, and 3D scenarios. This approach offers a boundary-only, PML-free alternative to traditional truncation techniques and shows strong potential for fast solvers and applications to layered media and waveguides.

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.

Figures (8)

• Figure 1: Perturbed half-space geometry
• Figure 2: Solution for data generated by a point source at $(1/10,-1)$. Left: the real part of the scattered field. Middle: the log (base 10) error computed using the analytic solution (in this case, the standard Helmholtz Green's function). In both cases the black line denotes the real part of the interface, and the blue line is its imaginary part. Right:$\log_{10}|\sigma|$ in part of $\Gamma_R.$
• Figure 3: Dependence of the error on the endpoints of the solution contour $\tilde{\Gamma}.$ The horizontal and vertical axis correspond to the real and imaginary parts of the endpoint of $\tilde{\Gamma}$ in $\Gamma_R$ ($\Im \,(\tilde{\Gamma})$ was chosen to be odd). The color denotes the $\log_{10}$ relative error.
• Figure 4: Solution of a Helmholtz Dirichlet problem with wavenumber $k=2\pi$ in a half-space. The data was generated from a plane source at $-45^{\rm o}$ to the interface. The reflected plane wave (at $45^{\rm o}$) is also subtracted. Left: real part of the scattered field. Right: Self-convergence plot of the error at $(-1,2)$ as a function of number of points used in the discretization.
• Figure 5: Solution due to a point source located at $(1/10, 10)$. Left: real part of the incident field. Middle: real part of the scattered field. Right: absolute value of the total field. In all three cases the black line denotes the real part of the interface, and the blue line is its imaginary part.

Theorems & Definitions (13)

• proof
• proof
• proof : Proof of \ref{['prop:Kdef']}
• proof
• proof : Proof of \ref{['lem:compact']}
• proof
• proof
• proof : Proof of \ref{['thm:inject']}
• proof : Proof of \ref{['thm:invert']}
• proof : Proof of \ref{['thm:main']}