In this paper, we present a multiscale framework for solving the Helmholtz equation in heterogeneous media without scale separation and in the high frequency regime where the wavenumber $k$ can be large. The main innovation is that our methods achieve a nearly exponential rate of convergence with respect to the computational degrees of freedom, using a coarse grid of mesh size $O(1/k)$ without suffering from the well-known pollution effect. The key idea is a coarse-fine scale decomposition of the solution space that adapts to the media property and wavenumber; this decomposition is inspired by the multiscale finite element method. We show that the coarse part is of low complexity in the sense that it can be approximated with a nearly exponential rate of convergence via local basis functions, while the fine part is local such that it can be computed efficiently using the local information of the right hand side. The combination of the two parts yields the overall nearly exponential rate of convergence. We demonstrate the effectiveness of our methods theoretically and numerically; an exponential rate of convergence is consistently observed and confirmed. In addition, we observe the robustness of our methods regarding the high contrast in the media numerically.
翻译:在本文中,我们提出了一个多尺度框架,用以在不进行比例分离的多种媒体和高频系统中解决Helmholtz方程式,而无需进行比例分解;这种分解是由多尺度的有限要素法启发的。我们指出,粗糙部分的复杂程度较低,因为通过本地基功能,可以接近于接近指数的趋同速度,而精细部分则是局部的,因此可以利用右手侧的当地信息进行高效率的计算。这两个部分的结合产生了总体接近指数的趋同速度。我们从理论上和数字上展示了我们的方法的有效性;趋同的指数率一直得到观察和确认。此外,我们观察了我们在数字媒体中高对比度方法的稳健性。