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.

我们通过分形平面屏幕研究了\（{\ mathbb {R}} ^ n \）（\（n = 2,3 \））中的时间谐波散射的边界元方法，假设该分形平面屏幕是一个非空有界子集超平面的\（\ Gamma \）\（\ Gamma _ \ infty = {\ mathbb {R}} ^ {n-1} \ times \ {0 \} \）。我们考虑两种不同的情况：（i）\（\ Gamma \）是具有分形边界的\（\ Gamma _ \ infty \）的相对开放子集（例如，在\（n = 3 \ ））; （ii）\（\ Gamma \）是\（\ Gamma _ \ infty \）的紧凑分形子集，内部为空（例如，在\ {n = 3 \的情况下为Sierpinski三角形））。在这两种情况下，我们的数值模拟策略都涉及用一系列更平滑的“预分形”屏幕逼近分形屏幕\（\ Gamma \），为此，我们使用离散化关联的第一类边界积分方程的边界元素方法来计算散射场。我们使用Mosco收敛框架证明了网格尺寸的充分条件，可确保收敛到极限分形解。我们还提供了数值示例来说明我们的理论结果。