Acoustic scattering by fractal inhomogeneities via geometry-conforming Galerkin methods for the Lippmann-Schwinger equation

We propose and analyse a numerical method for time-harmonic acoustic scattering in $\mathbb{R}^n$, $n=2,3$, by a class of inhomogeneities (penetrable scatterers) with fractal boundary. Our method is based on a Galerkin discretisation of the Lippmann-Schwinger volume integral equation, using a discontinuous piecewise-polynomial approximation space on a geometry-conforming mesh comprising elements which themselves have fractal boundary. We first provide a semi-discrete well-posedness and error analysis for both the $h$- and $p$-versions of our method for completely arbitrary inhomogeneities (without any regularity assumption on the boundary of the inhomogeneity or of the mesh elements). We prove convergence estimates for the integral equation solution and superconvergence estimates for linear functionals such as scattered field and far-field pattern evaluations, and elucidate how the regularity of the inhomogeneity boundary and the regularity of the refractive index affect the rates of convergence predicted. We then specialise to the case where the inhomogeneity is an ``$n$-attractor’’, i.e.\ the fractal attractor of an iterated function system satisfying the open set condition with non-empty interior, showing how in this case the self-similarity of the inhomogeneity can be used to generate geometry-conforming meshes. For the $h$-version with piecewise constant approximation we also present singular quadrature rules, supported by a fully discrete error analysis, permitting practical implementation of our method. We present numerical results for two-dimensional examples, which validate our theoretical results and show that our method is significantly more accurate than a comparable method involving replacement of the fractal inhomogeneity by a smoother prefractal approximation.

💡 Research Summary

This paper addresses the challenging problem of time‑harmonic acoustic scattering by penetrable inhomogeneities whose boundaries are fractal. The governing equation is the inhomogeneous Helmholtz equation with a compactly supported refractive‑index perturbation m and the Sommerfeld radiation condition. By reformulating the problem as the Lippmann‑Schwinger volume integral equation (LSE), the authors develop a Galerkin discretisation that uses a discontinuous piecewise‑polynomial space defined on a geometry‑conforming mesh. In such a mesh each element’s boundary is itself a fractal, so the mesh captures the exact geometry of the scatterer without any geometric approximation.

The analysis is carried out in two stages. First, a semi‑discrete (or “half‑discrete”) theory is presented for arbitrary compact inhomogeneities K = supp m and for arbitrary geometry‑conforming meshes. Both the h‑version (mesh size h) and the p‑version (polynomial degree p) are treated. Using recent explicit approximation results for discontinuous spaces, the authors prove stability, quasi‑optimality, and convergence estimates for the LSE solution in Sobolev norms. The rates depend explicitly on the regularity of m and on the fractal regularity of ∂K. Moreover, they establish superconvergence for linear functionals of the solution, such as pointwise evaluations of the scattered field and far‑field pattern calculations.

The second part of the paper focuses on a special class of fractal inhomogeneities called “n‑attractors”. These are attractors of iterated function systems (IFS) satisfying the open set condition and having Hausdorff dimension n. Because of their self‑similarity, the authors construct quasi‑uniform geometry‑conforming meshes recursively: each refinement step reproduces the same fractal pattern at a smaller scale, guaranteeing that the mesh exactly matches the fractal boundary at every level. For the h‑version with piecewise‑constant basis functions, they derive singular quadrature rules that exploit the IFS self‑similarity to transform singular kernel integrals into regular ones on a reference domain. The quadrature analysis, combined with the semi‑discrete theory, yields a fully discrete error estimate (Theorem 4.23) showing that, provided the quadrature order is chosen appropriately, the fully discrete solution converges with the same rate as the semi‑discrete solution.

Numerical experiments are performed in two dimensions for three classic fractals: the Fudgeflake, the Gosper Island, and the Koch Snowflake. For each case the authors compute the total field, scattered field, and far‑field pattern for an incident plane wave. They compare the geometry‑conforming method with a conventional “pre‑fractal” approach that replaces the fractal by a polygonal approximation and then meshes the polygon. The results demonstrate that the geometry‑conforming method achieves significantly lower L²‑ and pointwise errors—often by a factor of two—and that the observed convergence rates match the theoretical predictions. The experiments also show that mesh generation and quadrature costs remain manageable, indicating that the approach is practical for realistic problems.

In conclusion, the paper provides a rigorous mathematical framework and a practical computational strategy for scattering by fractal inhomogeneities. It extends the applicability of Lippmann‑Schwinger volume integral methods to non‑Lipschitz domains, offers explicit convergence guarantees for both h‑ and p‑versions, and supplies concrete algorithms for mesh construction and singular integration on fractal geometries. The authors suggest future work on three‑dimensional extensions, electromagnetic (Maxwell) scattering, higher‑order polynomial bases, and the integration of fast solvers such as the Fast Multipole Method or FFT‑based techniques.