Staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids.

在本文中，我們為了三維空間的馬克士威方程組(Maxwell’s equation)制定和分析了一套新種類的交錯間斷伽遼金(discontinuous Galerkin)方法，同時考慮了時間依賴性和時間諧波的馬克士威方程組。我們用了空間離散上交錯笛卡兒網格，這種方法具有許多良好的性質。首先，我們的方法所得出的數值解保留了電磁能量，並自動符合了高斯定律的離散版本。第二，質量矩陣是對角矩陣，從而時間推進是顯式和非常有效的。第三，我們的方法是高階準確，最佳收斂性在這裏會被嚴格地證明。第四，基於笛卡兒網格，它也很容易被執行，並可視為是典型的Yee’s Scheme的以及四邊形的邊有限元的推廣。最後，超收斂結果也會在這裏被證明。在本文中，我們還提供了幾個數值結果驗證了理論的陳述。我們計算了時間依賴性和時間諧波的馬克士威方程組數值收斂結果。此外，我們計算時間諧波馬克士威方程組特徵值問題的數值特徵值，並與理論特徵值比較結果。最後，完美匹配層(Perfect Matching Layer)吸收邊界的問題也有實行其數值結果。We develop and analyze a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations in this paper. Both time-dependent and time-harmonic Maxwell’s equations are considered. The spatial discretization is based on staggered Cartesian grids which possess many good properties. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Second, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Third, our method is high order accurate and the optimal order of convergence is rigorously proved. Fourth, it is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Lastly, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved.In this paper, we also provide several numerical results to verify the theoretical statements. We compute the numerical convergence order using L2-norm and discrete-norm respectively for both the time-dependent and time-harmonic Maxwell’s equations. Also, we compute the numerical eigenvalues for the time-harmonic eigenvalue problem and compare the result with the theoretical eigenvalues. Lastly, applications to problems in unbounded domains with the use of PML are also presented.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Yu, Tang Fei.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 46-49).Abstracts also in Chinese.Chapter 1 --- Introduction and Model Problems --- p.1Chapter 2 --- Staggered DG Spaces --- p.4Chapter 2.1 --- Review on Gauss-Radau and Gaussisan points --- p.5Chapter 2.2 --- Basis functions --- p.6Chapter 2.3 --- Finite Elements space --- p.7Chapter 3 --- Method derivation --- p.14Chapter 3.1 --- Method --- p.14Chapter 3.2 --- Time discretization --- p.17Chapter 4 --- Energy conservation and Discrete Gauss law --- p.19Chapter 4.1 --- Energy conservation --- p.19Chapter 4.2 --- Discrete Gauss law --- p.22Chapter 5 --- Error analysis --- p.24Chapter 6 --- Numerical examples --- p.29Chapter 6.1 --- Convergence tests --- p.30Chapter 6.2 --- Diffraction by a perfectly conducting object --- p.30Chapter 6.3 --- Perfectly matched layers --- p.37Chapter 7 --- Time Harmonic Maxwell’s equations --- p.40Chapter 7.1 --- Model Problems --- p.40Chapter 7.2 --- Numerical examples --- p.40Chapter 7.2.1 --- Convergence tests --- p.41Chapter 7.2.2 --- Eigenvalues tests --- p.41Chapter 8 --- Conclusion --- p.45Bibliography --- p.4

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