PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface

TL;DR

This paper develops a high-accuracy boundary integral equation solver combined with PML for wave scattering by a defected periodic surface, proving convergence and exponential decay of solutions, and demonstrating efficiency through numerical experiments.

Contribution

It introduces a novel PML-based BIE method with rigorous convergence analysis for wave scattering on defected periodic surfaces, including the use of Riccati equations for efficient computation.

Findings

PML solution converges linearly to the true solution within the strip.

Neumann-marching operators are contracting, leading to exponential decay of the PML solution.

Numerical experiments confirm exponential convergence of the PML solution in subdomains.

Abstract

This paper studies the PML method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges linearly to the true solution in the physical subregion of the strip with the PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem onto the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot exponentially converge to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip.