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
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328615 |
| Date | January 2012 |
| Contributors | Leung, Wing Tat., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (66 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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