亥姆霍茲方程(Helmholtz equation)是描述許多物理現象(如散射和波傳播)的數學 模型之一，使用數值方法去求解亥姆霍茲方程存在著一些困難。首先，缺乏魯棒 性被稱為污染效應。其次，當在無邊界域的外域中定義問題時，很難找到一個有 效的迭代求解器，隨著波數的增加時，這些迭代求解器可以收斂並且得到少量的 迭代次數。本文提出了一個多尺度有限元方法(MsFEM)的新框架作為這類問題 的迭代求解器。該方法通過所謂的自適應氣泡函數來改進，使得該方法被稱為具 有自適應氣泡函數富集的多尺度有限元方法(MsFEM bub)。各種波數的數值實 驗中表現出該方法的穩健性和效率。

The Helmholtz equation is one of mathematical model to describe many physical phenomena, such as scattering and wave propagation. There are difficulties of solving Helmholtz equation numerically. First, the lack of robustness which is called pollution effect. Second, when the problem defined in an unbounded domain exterior domain, it is hard to find an efficient iterative solver that converge with a few number of iterations as the wavenumber increasing. This thesis presents a new framework of multiscale finite element method (MsFEM) as an iterative solver for such problems. The method is improved by so-called adaptive bubble function such that the method is called multiscale finite element method with adaptive bubble function enrichment (MsFEM bub). Numerical experiments for various wavenumbers indicate the robustness and the efficiency of the method.

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