Integral Equation Methods for Scattering by Multifractal Obstacles

TL;DR

This paper extends integral equation methods for acoustic scattering to multifractal obstacles with spatially varying fractal dimensions, providing new theoretical insights and convergence results for numerical schemes.

Contribution

It generalizes existing models to multifractal scatterers, interprets the operator equation as a trace space problem, and establishes convergence criteria for Galerkin methods on complex measures.

Findings

Operator equation interpreted as between trace space and dual

Integral equation equivalence under measure conditions

Convergence of Galerkin methods depends on density of smooth functions

Abstract

Caetano et al. (Proc. R. Soc. A. 481:20230650, 2025) have proposed a formulation for sound-soft acoustic scattering by a compact scatterer O $\subset$ Rn, in which the scattered field is represented as an acoustic Newtonian potential whose density is the solution of an operator equation on a compact set $\Gamma$ $\subset$ O. In the case that $\Gamma$ is Ahlfors-David d-regular (a d-set), for some d $\in$ (n--2, n], they show, moreover, that the operator equation can be interpreted as an integral equation, the integration with respect to d-dimensional Hausdorff measure, and present a convergent Galerkin scheme for numerical computation. In this paper we make a substantial extension of these results so that they apply to more realistic fractal scatterers that are multifractal, in the sense that they have spatially varying fractal dimension. Firstly, we provide, inspired by Claret et al. (J. Math. Pures Appl. 212:103888, 2026), an interpretation of this operator equation as an equation between a trace space on $\Gamma$ and its dual, and, in many cases, relate the density to a notion of the normal derivative of the scattered field on $\Gamma$. Secondly, we show that the operator equation is equivalent to an integral equation on $\Gamma$ whenever $\Gamma$ is the support of a Radon measure $\mu$ such that: (i) the trace operator from H1(Rn) to L2($\Gamma$, $\mu$) is continuous and; (ii) certain canonical singular integrals with respect to $\mu$ are finite; and we characterise a large class of measures for which (i) and (ii) hold. Finally, we show that Galerkin methods based on finite element subspaces of L2($\Gamma$, $\mu$) are convergent if and only if, additionally, C$\infty$\_0 (Rn\$\Gamma$) is dense in the kernel of the trace operator. These results apply, in particular, if $\Gamma$ is a finite union of d-sets with different values of d. In the case that each d-set is the attractor of an iterated function system of contracting similarities, we establish rates of convergence for the Galerkin method.