通过几何适应的伽辽金方法求解 fractal 不均匀体的声散射
Acoustic scattering by fractal inhomogeneities via geometry-conforming Galerkin methods for the Lippmann-Schwinger equation
AI总结：
本文提出了一种基于几何适应伽辽金方法的数值方法，用于求解具有分形边界的不均匀体的声散射问题，通过理论分析和数值实验验证了该方法的高精度和有效性。
摘要：
我们提出并分析了一种用于求解 $\mathbb{R}^n$（$n=2,3$）中时谐声散射的数值方法，该方法用于由具有分形边界的不均匀体（穿透散射体）引起的散射。我们的方法基于对利普曼-施温格体积积分方程的伽辽金离散化，使用在由具有分形边界的元素组成的几何适应网格上进行的不连续分段多项式近似空间。我们首先为我们的方法的 $h$-版本和 $p$-版本提供了半离散的适定性和误差分析，适用于完全任意的不均匀体（不假设不均匀体边界或网格元素的正则性）。我们证明了积分方程解的收敛估计以及线性泛函（如散射场和远场模式评估）的超收敛估计，并阐明了不均匀体边界和折射率的正则性如何影响预测的收敛率。然后我们专门讨论不均匀体为“$n$-吸引子”的情况，即满足开放集条件且具有非空内部的迭代函数系统的分形吸引子，展示了在这种情况下如何利用不均匀体的自相似性来生成几何适应的网格。对于使用分段常数近似的 $h$-版本，我们还提出了奇异求积规则，这些规则由完全离散误差分析支持，使得我们的方法能够实际实施。我们为二维示例提供了数值结果，这些结果验证了我们的理论结果，并表明我们的方法比使用将分形不均匀体替换为更光滑的预分形近似的方法更准确。
英文摘要：
We propose and analyse a numerical method for time-harmonic acoustic scattering in $\mathbb{R}^n$, $n=2,3$, by a class of inhomogeneities (penetrable scatterers) with fractal boundary. Our method is based on a Galerkin discretisation of the Lippmann-Schwinger volume integral equation, using a discontinuous piecewise-polynomial approximation space on a geometry-conforming mesh comprising elements which themselves have fractal boundary. We first provide a semi-discrete well-posedness and error analysis for both the $h$- and $p$-versions of our method for completely arbitrary inhomogeneities (without any regularity assumption on the boundary of the inhomogeneity or of the mesh elements). We prove convergence estimates for the integral equation solution and superconvergence estimates for linear functionals such as scattered field and far-field pattern evaluations, and elucidate how the regularity of the inhomogeneity boundary and the regularity of the refractive index affect the rates of convergence predicted. We then specialise to the case where the inhomogeneity is an ``$n$-attractor'', i.e.\ the fractal attractor of an iterated function system satisfying the open set condition with non-empty interior, showing how in this case the self-similarity of the inhomogeneity can be used to generate geometry-conforming meshes. For the $h$-version with piecewise constant approximation we also present singular quadrature rules, supported by a fully discrete error analysis, permitting practical implementation of our method. We present numerical results for two-dimensional examples, which validate our theoretical results and show that our method is significantly more accurate than a comparable method involving replacement of the fractal inhomogeneity by a smoother prefractal approximation.
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