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Acoustic scattering by impedance screens/cracks with fractal boundary: Well-posedness analysis and boundary element approximation
Mathematical Models and Methods in Applied Sciences ( IF 3.5 ) Pub Date : 2022-02-07 , DOI: 10.1142/s0218202522500075
J. Bannister 1 , A. Gibbs 1 , D. P. Hewett 1
Affiliation  

We study time-harmonic scattering in 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 Γ = n1 ×{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 Γ 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.

中文翻译:

具有分形边界的阻抗屏/裂缝的声散射:适定性分析和边界元逼近

我们研究时谐散射n(n = 2, 3)通过平面屏幕(线性弹性上下文中的“裂缝”),假设为非空有界相对开放子集Γ超平面的Γ = n-1 ×{0}, 在其上施加阻抗 (Robin) 边界条件。与以往的研究相比,Γ可以有任意粗糙的(可能是分形的)边界。为这样的人获得适姿性Γ我们展示了标准阻抗边值问题及其相关的边界积分方程组必须如何补充额外的解规律性条件,这些条件在以下情况下自动成立Γ是光滑的。我们证明了边界积分算子的相关系统在适当的函数空间设置中是紧扰动的,加强了先前的结果。这允许使用 Mosco 收敛来证明更平滑的“预分形”屏幕上的边界元近似收敛到分形屏幕上的极限解。我们提供了伴随的数值结果,验证了我们的理论收敛结果,用于科赫雪花和方形雪花的三维散射。
更新日期:2022-02-07
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