A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

TL;DR

This paper develops and analyzes a Hausdorff-measure boundary element method for acoustic scattering by fractal planar screens, formulating the problem in $H^{-1/2}_{\Gamma}$ with basis functions supported on the fractal set and integrating against the Hausdorff measure $\mathcal{H}^d$. It proves Galerkin convergence for compact $d$-sets with $n-1<d\le n$, and, for disjoint IFS attractors, establishes best-approximation and Galerkin error bounds, as well as inverse estimates and fully discrete convergence via Hausdorff quadrature. The authors design quadrature rules exploiting self-similarity, derive Strang-type perturbation results for fully discrete schemes, and validate the theory with extensive numerical experiments on Cantor sets and Cantor dusts, including comparisons with prefractal BEM and scenarios beyond the current theory. A central unresolved issue is Conjecture on the invertibility of the operator $\mathbb{S}$ on a range of trace spaces; resolving it would solidify convergence rates and enable broader applicability, including Neumann/impedance problems and non-disjoint fractal screens. Overall, the work provides a rigorous, practical framework for acoustics involving fractal screens and demonstrates that fractal geometry can be naturally integrated into BEM through Hausdorff-measure discretization.

Abstract

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this ``Hausdorff BEM'' for acoustic scattering in $\mathbb{R}^{n+1}$ ($n=1,2$) when the scatterer, assumed to be a compact subset of $\mathbb{R}^n\times\{0\}$, is a $d$-set for some $d\in (n-1,n]$, so that, in particular, the scatterer has Hausdorff dimension $d$. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in $\mathbb{R}^2$ by Cantor sets, and in $\mathbb{R}^3$ by Cantor dusts.

Figures (18)

• Figure 1: The first five standard prefractal approximations, $\Gamma_0,\ldots,\Gamma_4$, to the middle-third Cantor dust $\Gamma$, defined by $\Gamma_0:=[0,1]^2$, $\Gamma_n:=s(\Gamma_{n-1})$, $n\in \mathbb{N}$, where $s$ is defined by \ref{['eq:fixedfirst']} and \ref{['eq:CD_IFS']} with $M=4$ and $\rho=1/3$.
• Figure 2: The main function spaces introduced in § \ref{['sec:FunctionSpaces']} and their relations. Here $\Gamma\subset\mathbb{R}^n$ is a $d$-set with $0<d\leq n$, $t=s-\frac{n-d}{2}\in(0,1)$, both arrows represent unitary isomorphisms. Where we write $A\subset B$, the function space $A$ is densely and continuously embedded in $B$. Where $\cap$ separates $A$ and $B$ vertically, the space $A$ is a closed subspace of $B$.
• Figure 3: Illustration of the sets $E_{\bm}$, $\bm\in I_\ell$, $\ell=0,1,2$, with $E=[0,1]$, for the IFS $s_1(x)=0.4x$, $s_2(x)=0.15x+0.5$, $s_3(x)=0.25x+0.75$, associated with a Cantor-type set with $M=3$.
• Figure 4: Graphs of the orthonormal basis functions $\psi^1,\psi^2$ of $\mathbb{W}_1\ominus\mathbb{W}_0$ (left) and $\psi^m_\bm$, $\bm\in I_1$, of $\mathbb{W}_2\ominus\mathbb{W}_1$ (right) for the IFS of Figure \ref{['fig:Basis']}. The black lines are the components of the attractor $\Gamma$. Where the values of $\psi^1$, $\psi^2$, and $\psi^n_\bm$ on $\Gamma$ are not shown explicitly, the values are zero, i.e. the graphs coincide with the black lines.
• Figure 5: Schema of relevant function spaces and operators for $s=1/2$, $t=t_d:={1/2}-\frac{n-d}{2}$.

Theorems & Definitions (70)

• Lemma 2.1
• Remark 2.2
• Corollary 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Theorem 2.7
• Theorem 3.1: Jonsson98