Computing scattering resonances of rough obstacles

TL;DR

This work develops a rigorous, computable framework for extracting scattering resonances of the Laplacian on exterior domains with rough boundaries, including fractal boundaries. It combines a domain-decomposition approach, NtD operators, spectral expansions, and FEM to build a finite, well-conditioned operator $T(k)$ whose zeros approximate resonances; convergence is proven under Mosco convergence and controlled boundary polygonal approximations, with error terms tied to matrix truncation, sum truncation, and FEM discretisation. The authors prove a solvability-complexity result placing the problem in the SCI class $\Delta_2^A$, ensuring convergence of a sequence of arithmetic algorithms to the true resonance set, and they demonstrate the method on disks, Koch snowflakes, and filled Julia sets with MATLAB experiments. The numerical pipeline relies on zeros of an analytic determinant $g_n(k)=\det T_n(k)$, allowing robust error control via analytic function-zero techniques and the argument principle. Overall, the paper provides a practical, provably convergent route to compute resonances for highly irregular obstacles, enabling exploration of fractal-boundary scattering phenomena in two dimensions.

Abstract

This paper is concerned with the numerical computation of scattering resonances of the Laplacian for Dirichlet obstacles with rough boundary. We prove that under mild geometric assumptions on the obstacle there exists an algorithm whose output is guaranteed to converge to the set of resonances of the problem. The result is formulated using the framework of Solvability Complexity Indices. The proof is constructive and provides an efficient numerical method. The algorithm is based on a combination of a Glazman decomposition, a polygonal approximation of the obstacle and a finite element method. Our result applies in particular to obstacles with fractal boundary, such as the Koch Snowflake and certain filled Julia sets. Finally, we provide numerical experiments in MATLAB for a range of interesting obstacle domains.

Figures (13)

• Figure 1: Sketch of the definitions of $\Omega$ and $\Omega_{\text{out}}$. The dashed line in the left figure indicates $B_{X-1}(0)$.
• Figure 2: Sketch of a pixelation approximation (left) and pre-fractal approximation (right) of $\Omega$.
• Figure 3: Triangulations of an annulus with an inner radius of 0.5 and outer radius of 0.7 for two different values of the mesh parameter $h$. Left: $h=0.05$, right: $h=0.01$.
• Figure 4: Resonances of $U$ in the complex plane. Left: Computed directly from Hankel functions with high accuracy. Right: Contour plot of $\log|\det(T_n(k))|$, computed as described in Section , together with its zeros (red dots). The reference point
• Figure 5: Left: diagonal elements of $|T_n|$ for $h=0.08$, $N=6$. The red dots mark its minima, which we interpret as the optimal size of the matrix $T_n$. Right: square of optimal value of $N$ (determined as in Section \ref{['sec:optimal_N_heuristic']}), plotted against $h^{-1}$.

Theorems & Definitions (60)

• Definition 2.1
• Definition 2.2: Arithmetic algorithm
• Theorem 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Lemma 2.7
• Remark 2.8
• Theorem 2.13
• Example 1: Pixelated domains