Error Estimates of Discontinuous Galerkin Methods for Elliptic Problems
- Abstract
- 본 학위논문에서는 과학과 공학에서 다양한 문제를 설명하는 동차 디리클레 경계조건을 가지는 1차원과 2차원 타원형문제들에 대해서 내부 페널티를 가지는 불연속 갈레르킨 방법에 의한 근사해를 구성하고, 근사해의 존재성을 보였다. 또한 근사해의 에너지 노름과 노름에 대한 오차가 최적이 됨을 입증하였다.
그리고 1차원과 2차원 타원형문제에 고안한 내부 페널티를 가지는 불연속 갈레르킨 방법을 적용하여 근사해의 노름에 대한 이론적 결과를 수치적으로 입증하였다.
For one and two dimensional elliptic problems with homogeneous Dirichlet boundary condition arising in the various areas of sciences and engineering, we construct approximate solutions of the problems based on the discontinuous Galerkin methods with an interior penalty and show the existence of approximate solutions. And we establish the optimality of error estimates in the energy and norms for the discontinuous Galerkin approximations.
Adapting our discontinuous Galerkin methods with an interior penalty to one and two dimensional elliptic problems, we verify numerically the optimal results of error estimates in the norm for the discontinuous Galerkin approximations.
- Issued Date
- 2007
- Awarded Date
- 2007. 2
- Type
- Dissertation
- Keyword
- discontinuous Galerkin methods elliptic problems Error Estimates Discontinuous Galerkin Methods Elliptic Problems
- Publisher
- 부경대학교 대학원
- Alternative Author(s)
- Lee, Min-A
- Affiliation
- 부경대학교 대학원
- Department
- 대학원 응용수학과
- Advisor
- 신준용
- Table Of Contents
- Chapter 1 Introduction = 1
Chapter 2 Preliminaries = 3
2.1 Function Spaces = 3
2.2 Trace Inequalities = 5
2.3 Lax-Milgram Theorem = 7
2.4 Approximation Properties = 8
2.5 Generalized Minimum Residual Method = 9
Chapter 3 One Dimesional Elliptic Problems = 13
3.1 Introduction = 13
3.2 Notations = 14
3.3 A Discontinuous Weak Formulation = 17
3.4 DGFEMs with an Inerior Penalty = 19
3.5 Error Estimates = 26
Chapter 4 Two Dimesional Elliptic Problems = 33
4.1 Introduction = 33
4.2 Notations = 34
4.3 A Discontinuous Weak Formulation = 37
4.4 DGFEMs with an Inerior Penalty = 40
4.5 Error Estimates = 47
Chapter 5 Numerical Experiments = 55
5.1 One Dimensional Elliptic Problems = 55
5.2 Two Dimensional Elliptic Problems = 78
References = 88
- Degree
- Doctor
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