论文标题
带有分形边界的阻抗屏幕/裂纹的声学散射：适当的分析和边界元素近似
Acoustic scattering by impedance screens/cracks with fractal boundary: well-posedness analysis and boundary element approximation
论文作者
论文摘要
我们在$ \ mathbb {r}^n $（$ n = 2,3 $）中研究平面屏幕（线性弹性中的“裂缝”）中的时间谐波散射，假定是一种非空的相对开放的子集$γ$，它是增生的$ \ \ mathbb {r}^n-1} $ c {n-1} $ c {n-1} $ \ c {0 （罗宾）施加边界条件。与以前的研究相反，$γ$可以任意具有粗糙的（可能是分形）边界。为了获得此类$γ$的适当性，我们显示标准阻抗边界值问题及其相关的边界积分方程系统如何补充其他解决方案规则性条件，当$ \partialγ$平滑时，它们会自动保持。我们表明，边界积分运算符的相关系统在适当的功能空间设置中被紧凑地扰动强制性，从而增强了先前的结果。这允许使用MOSCO收敛来证明在分形屏幕上更平滑的“预三十叶”屏幕上边界元素近似的收敛。我们提出了伴随的数值结果，以验证我们的理论收敛结果，以通过科赫雪花和方形雪花进行三维散射。
We study time-harmonic scattering in $\mathbb{R}^n$ ($n=2,3$) by a planar screen (a "crack" in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $Γ$ of the hyperplane $\mathbb{R}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $Γ$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $Γ$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when $\partialΓ$ is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother "prefractal" screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.
