離散旋度算子的特徵分解及其在馬克斯威爾方程之應用

Abstract

在這篇文章，我們考慮從馬克斯威爾方程所推導出的線性系統及特徵值問題，並為其設計了一系列的快速演算法。我們最開始研究了光子晶體問題，光子晶體的數學模型是馬克斯威爾的特例，在某些簡化假設之下，這個特例被稱為時間諧波馬克斯威爾方程，我們參考了一些光子晶體相關的文獻，並且學習了一些馬克斯威爾方程的離散方法。最後我們採用了 Yee’s Scheme 作為我們的主要方法。根據 Yee’s Scheme方法，我們能將剛才提及的時間諧波馬克斯威爾方程轉換成一個推廣的特徵值問題，接著我們開始研究這個推廣特徵值問題，經過了一些努力，我們找到了一種雙旋度算子的顯式特徵分解，再加上一些矩陣計算技巧，我們能讓整體問題的計算速度有了顯著的提升，這樣的方法被我們稱為零空間免除法。更進一步，我們還構造了單旋度算子的奇異值分解來處理一些對掌性材料及電漿數值模擬問題。最後我們推廣了這些分解的結果，使其可以應用在不同的邊界條件上，像是完美匹配層邊界，Dirichlet 邊界條件，以及擬週期邊界條件等等。

In this article, we mainly consider how to design some fast algorithms for some eigenvalue problems and linear systems which are derived from Maxwell's equations. We first started from the issue of photonic crystal, the mathematical model of photonic crystal is a special case of Maxwell equations under some simplifying assumptions,which is called time harmonic Maxwell's equations. We refer to some photonic crystal research papers and learn some discretization methods of Maxwell's equations. Finally, we mainly used Yee's Scheme method, by applying Yee's Scheme method, the time harmonic Maxwell's equations will be transformed into a generalized eigenvalue problem, so we began to study the generalized eigenvalue problem. After some effort, we found an explicit eigen-decomposition of the double curl discrete matrix and used some matrix computation techniques to accelerate computation speed significantly, we called this algorithm null space free method. Moreover, we construct singular value decomposition of a single curl from the previous eigen-decomposition, then we used such decomposition to solve the chiral medium and plasma numerical simulation problems. Finally, we generalized the results of these decompositions, so that these techniques can be used under different boundary conditions, such as perfect matching boundary condition, Dirichlet boundary condition and quasi-periodic boundary condition, etc.