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Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Jan. 13, 2009). PDF text: v, 128 p. : col. ill. ; 2 Mb. UMI publication number: AAT 3315327. Includes bibliographical references. Also available in microfilm and microfiche formats.

Inaug.-Diss.--Rheinische Friedrich-Wilhelms-Universität zu Bonn, 1986. / Publication date on cover: 1987. Includes bibliographical references (p. 119-121).

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Mary F. Wheeler. Vita. Includes bibliographical references.

This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is reduced, yet the convergence is recovered regardless of parity.

擴散和對流擴散問題在高度異質性介質以及對流佔優擴散問題一直是一個具挑戰性的問題。眾所周知，對這些問題的數值計算需要相當數量的計算機內存和時間。然而，這些問題的解決方案通常包含一個粗糙的組成部分，它通常是我們感興趣的的量，而這個量一般可以用一個小數目的自由度代表。 / 現今有許多不同方法的目標是在計算粗糙的組件而不計算解的全部細節。在這篇論文中，我們將使用多尺度間斷伽遼金(Galerkin)方法來解決這問題。我們的方法是一種內部處罰間斷伽遼金方法。內部處罰間斷伽遼金方法已被證明是有效和準確的方法用來計算偏微分方程的數值解。我們的方法的一個顯著特點是解空間包含兩個部分，其中一個是傳統的多項式空間，另一個是計算粗糙的組件必不可少的包含分格結構的多尺度空間。我們在這篇論文中證明了該方法的穩定性。此外計算結果中表明，該方法能夠準確地捕捉高度異質性介質中的問題以及邊界和內部層對流佔優問題的解。 / Diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem has been a challenge problem for a long time. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. / There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. In this thesis, we will investigate a multisacle discontinuous Galerkin method to solve the problems. This method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Leung, Wing Tat. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 63-66). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Overview --- p.6 / Chapter 1.2 --- Strong Formulation --- p.8 / Chapter 1.3 --- Weak Formulation --- p.8 / Chapter 1.4 --- Finite Element Method --- p.10 / Chapter 2 --- The discontinuous Galerkin (DG) method --- p.12 / Chapter 2.1 --- Introducion --- p.12 / Chapter 2.2 --- Model problem --- p.13 / Chapter 2.3 --- Interior Penalty Method --- p.14 / Chapter 3 --- The Multiscale Method --- p.18 / Chapter 3.1 --- Introducion --- p.18 / Chapter 3.2 --- Multiscale finite element method --- p.19 / Chapter 3.3 --- Homogenization --- p.20 / Chapter 3.4 --- Convergence --- p.23 / Chapter 3.4.1 --- Convergence for H < ε --- p.23 / Chapter 3.4.2 --- Convergence for H > ε --- p.24 / Chapter 3.5 --- Singularly perturbed convection-diffusion problem --- p.25 / Chapter 4 --- The multiscale-enhanced IPDG method --- p.27 / Chapter 4.1 --- Method description --- p.27 / Chapter 4.2 --- Stability --- p.31 / Chapter 4.3 --- Boundedness --- p.32 / Chapter 4.4 --- Convergence for elliptic problem --- p.37 / Chapter 5 --- Numerical Result --- p.46 / Chapter 5.1 --- Accuracy and convergence tests --- p.46 / Chapter 5.2 --- Some other cases with smooth coefficient --- p.49 / Chapter 5.3 --- Performance with internal or boundary layers --- p.52 / Chapter 5.4 --- Time-dependent problems --- p.55 / Chapter 5.5 --- Performance with more general media --- p.59

Lee, Chak Shing. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Model Problems --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- The curl-curl operator --- p.2 / Chapter 1.3 --- The convection-diffusion equation --- p.6 / Chapter 2 --- Staggered DG method for the Curl-Curl operator --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Discontinuous Galerkin discretization --- p.8 / Chapter 2.3 --- Stability for aligned fields --- p.14 / Chapter 2.4 --- Error estimates --- p.17 / Chapter 2.5 --- Numerical experiments --- p.21 / Chapter 2.6 --- Concluding Remarks --- p.32 / Chapter 3 --- Staggered DG method for the convection-diffusion equation --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.2 --- Method description --- p.33 / Chapter 3.3 --- Preservation of physical structures --- p.38 / Chapter 3.4 --- Stability and convergence --- p.42 / Chapter 3.4.1 --- Static problem --- p.42 / Chapter 3.4.2 --- Time-dependent problem --- p.46 / Chapter 3.5 --- Fully discrete scheme --- p.49 / Chapter 3.6 --- Numerical examples --- p.55 / Chapter 3.6.1 --- The static problem --- p.55 / Chapter 3.6.2 --- Time dependent problem --- p.56 / Chapter 3.7 --- Concluding Remark --- p.59 / Bibliography --- p.60

by Chan Chung-Kei, Thomas. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 125-[129]). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Semiconductor Device Physics --- p.5 / Chapter 2.1 --- IC Design Methodology --- p.6 / Chapter 2.1.1 --- System Level --- p.7 / Chapter 2.1.2 --- Circuit Level --- p.7 / Chapter 2.1.3 --- Device Level --- p.8 / Chapter 2.1.4 --- Process Level --- p.8 / Chapter 2.2 --- Classification of Device Models --- p.8 / Chapter 2.2.1 --- Circuit Models --- p.9 / Chapter 2.2.2 --- Physical Models --- p.10 / Chapter 2.3 --- Classical Drift-Diffusion model --- p.13 / Chapter 2.3.1 --- Basic Governing Equations in Semiconductors --- p.13 / Chapter 2.3.2 --- Shockley-Read-Hall Recombination Statics --- p.15 / Chapter 2.3.3 --- Boundary Conditions --- p.18 / Chapter 2.4 --- pn Junction at equilibrium --- p.20 / Chapter 2.4.1 --- The depletion approximation --- p.23 / Chapter 2.4.2 --- Current-voltage Characteristics --- p.26 / Chapter 3 --- Iteration Scheme --- p.30 / Chapter 3.1 --- Gummel's iteration scheme --- p.31 / Chapter 3.2 --- Modified Gummel's iteration scheme --- p.35 / Chapter 3.3 --- Solution of Differential Equation --- p.38 / Chapter 3.3.1 --- Finite Difference Method --- p.38 / Chapter 3.3.2 --- Moment Method --- p.39 / Chapter 4 --- Theory of Wavelets --- p.43 / Chapter 4.1 --- Multi-resolution Analysis --- p.43 / Chapter 4.1.1 --- Example of MRA with Haar Wavelet --- p.46 / Chapter 4.2 --- Orthonormal basis of Wavelets --- p.52 / Chapter 4.3 --- Fast Wavelet Transform --- p.56 / Chapter 4.4 --- Wavelets on the interval --- p.62 / Chapter 5 --- Galerkin-Wavelet Method --- p.66 / Chapter 5.1 --- Wavelet-based Moment Methods --- p.67 / Chapter 5.1.1 --- Wavelet transform on the stiffness matrix --- p.67 / Chapter 5.1.2 --- Wavelets as basis functions --- p.68 / Chapter 5.2 --- Galerkin-Wavelet method --- p.69 / Chapter 5.2.1 --- Boundary Conditions --- p.73 / Chapter 5.2.2 --- Adaptive Scheme --- p.74 / Chapter 5.2.3 --- The Choice of Classes of Wavelet Bases --- p.76 / Chapter 6 --- Numerical Results --- p.80 / Chapter 6.1 --- Steady State Solution --- p.81 / Chapter 6.1.1 --- Daubechies Wavelet N = 2 --- p.82 / Chapter 6.1.2 --- Daubechies Wavelet N=5 --- p.84 / Chapter 6.1.3 --- Discussion on Daubechies wavelets N = 2 and N=5 --- p.86 / Chapter 6.2 --- Transient Solution --- p.91 / Chapter 6.3 --- Convergence --- p.99 / Chapter 7 --- Conclusion --- p.103 / Chapter A --- Derivation for steady state --- p.107 / Chapter A.1 --- Generalized Moll-Ross Relation --- p.107 / Chapter A.2 --- Linearization of PDEs --- p.110 / Chapter B --- Derivation for transient state --- p.113 / Chapter C --- Notation --- p.119 / Chapter D --- Elements in the Stiffness Matrix --- p.122 / Bibliography --- p.125

Radiative transfer theory describes the interaction of radiation with scattering and absorbing media. It has applications in neutron transport, atmospheric physics, heat transfer, molecular imaging, and others. In steady state, the radiative transfer equation is an integro-differential equation of five independent variables. This high dimensionality and presence of integral term present a serious challenge when trying to solve the equation numerically. Over the past 50 years, several techniques for solving the radiative transfer equation have been introduced. These include, but are certainly not limited to, Monte Carlo methods, discrete-ordinate methods, spherical harmonics methods, spectral methods, finite difference methods, and finite element methods. Methods involving discrete ordinates have received particular attention in the literature due to their relatively high accuracy, flexibility, and relatively low computational cost. In this thesis we present a discrete-ordinate discontinuous Galerkin method for solving the radiative transfer equation. In addition, we present a generalized Fokker-Planck equation that may be used to approximate the radiative transfer equation in certain circumstances. We provide well posedness results for this approximation, and introduce a discrete-ordinate discontinuous Galerkin method to approximate a solution. Theoretical error estimates are derived, and numerical examples demonstrating the efficacy of the methods are given.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 167-180). Available also in a digital version from Dissertation Abstracts.