光滑变系数高频散射问题的自适应有限元法
An adaptive finite element method for high-frequency scattering problems with smoothly varying coefficients
摘要：
介绍了一种求解时谐声散射问题的新的数值方法。主要的焦点是平面波散射的平滑变化的材料不均匀性。所提出的方法适用于任何频率$ω$，但特别是高频率的问题是有效的。它是基于一个时域的方法，由三个步骤组成：\emph{i)}计算一个合适的传入平面小波与紧凑的支持在传播方向; \emph{ii)}解决散射问题，在时域传入平面小波; \emph{iii)}重建的时间-调和解从时域解决方案通过傅里叶变换在时间。新方法的一个重要组成部分是一个前跟踪网格自适应算法解决问题的\emph{ii)}。通过利用波前的有限支持，这允许我们使达到给定精度所需的自由度的数量显著更少地依赖于频率$ω$。我们还提出了一种新的算法，用于计算傅里叶变换的\emph{iii)}，利用相应的自适应网格的自由度减少。数值算例表明，该方法的优点和事实，该方法也可以适用于外部源项，如点源和软声散射体。然而，所获得的效率在存在捕获模式的情况下是有限的。
英文摘要：
We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency $ω$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $ω$. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes. Numerical examples demonstrate the advantages of the proposed method and the fact that the method can also be applied with external source terms such as point sources and sound-soft scatterers. The gained efficiency, however, is limited in the presence of trapping modes.
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