We study boundary element methods for time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a fractal planar screen, assumed to be a non-empty bounded subset $\Gamma$ of the hyperplane $\Gamma_\infty=\mathbb{R}^{n-1}\times \{0\}$. We consider two distinct cases: (i) $\Gamma$ is a relatively open subset of $\Gamma_\infty$ with fractal boundary (e.g.\ the interior of the Koch snowflake in the case $n=3$); (ii) $\Gamma$ is a compact fractal subset of $\Gamma_\infty$ with empty interior (e.g.\ the Sierpinski triangle in the case $n=3$). In both cases our numerical simulation strategy involves approximating the fractal screen $\Gamma$ by a sequence of smoother "prefractal" screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. We also provide numerical examples illustrating our theoretical results.

翻译：我们研究的是以$mathb{R ⁇ n$(n=2,3美元)以折形平面屏幕进行时间和谐散射的边界元素方法。 我们研究的是两个不同的案例:(一)$\Gamma$是一个相对开放的子集,其中带有分形边界(例如,Koch雪花的内部值为$=3美元);(二)$\Gamma$是超平板机的非无界子集,其中含有$\Gamma_inffy}mathb{R ⁇ n-1 ⁇ _ ⁇ %0美元。我们考虑了两个不同的案例:(一)$\Gamma$是一个相对开放的子集成,其中含有$\Gamma_inffty}mathbright 美元,其中含有一个非空面平面平面平面平面平面平面屏幕(例如,Koch 雪花内部雪花的内部值为$\Gammamall 3美元);(二) $\Gamma$$Gamma$,其中我们用离散的离心平面平面图模型模型构建了离心平面的内框, 也提供了我们离心阵面的平面的内框的内框的内框。