The hybrid difference methods for convection-diffusion equations and interface problems
- 주제(키워드) hybrid difference method , convection-diffusion , interface problem
- 발행기관 아주대학교
- 지도교수 Youngmok Jeon
- 발행년도 2018
- 학위수여년월 2018. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/ajou/000000027781
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
In this study we aim to introduce and analyze the hybrid difference methods (HDMs) for convection-diffusion equations and the elliptic interface problems. The HDMs are composed of the finite difference approximation of the PDEs within cells (cell FD) and the finite difference approximation of the continuity of normal fluxes on the edges of cells (intercell FD). Numerical analysis of the HDMs is possible in a variational setting as other hybridized finite element methods. Moreover, the HDMs significantly reduce degrees of freedom and work well on non-uniform grids retaining convergence order. For convection dominated diffusion equations we propose two methods so-called the upwind hybrid difference method and the penalized upwind method. It is well known that the solutions of convection-diffusion equations typically contain interior and boundary layers and numerical solutions obtained by standard methods usually exhibit spurious and nonphysical oscillations. The upwind hybrid difference method is stable but numerical solution may contain oscillations due to the small diffusion term. By adding a penalty parameter in the intercell finite difference, which is oftenly chosen in the form of some power of the grid size, we obtain the penalized upwind method. Numerical experiments show that with an appropriate choice of the penalty parameter the penalized upwind method can give access to controlling oscillation as well as smearing of the numerical solutions. The hybrid difference approach can also be used to solve the elliptic interface problems. For that we introduce the interface hybrid difference (IHD) method. The domain is decomposed into cells so that each cell is contained exclusively in one of the subregions of the domain. The mesh does not required to be uniform and it is constructed to satisfy the deviation condition. The cells away from the interface are regular rectangles and the cells adjacent to the interface are trapezoids. In two dimensional case, a new intercell condition is introduced when the cell boundary coincides with the interface. If the interface is a polygon then the interface hybrid difference method is proved to have an optimal order of convergence in the energy norm. Several numerical experiments are presented to confirm the theoretical results.
more목차
Abstract i
List of Tables iv
List of Figures v
1 Introduction 1
2 Preliminaries 8
2.1 Notations ............................ 8
2.2 Function spaces......................... 9
2.3 Basic concepts in FEMs .................... 12
2.4 Useful formulas and inequalities................ 14
3 Convection-diffusion equations 15
3.1 Introduction........................... 15
3.1.1 Classical and weak solutions.............. 16
3.1.2 Boundary and interior layers. . . . . . . . . . . . . . 17
3.2 Numerical methods for convection diffusion equations . . . . 21
3.2.1 The upwind finite difference method . . . . . . . . . 21
3.2.2 The standard Galerkin method . . . . . . . . . . . . 21
3.2.3 The viscosity method.................. 23
3.2.4 The streamline diffusion method. . . . . . . . . . . . 24
4 The elliptic interface problems 27
4.1 Introduction........................... 27
4.2 The immersed interface method (IIM) . . . . . . . . . . . . 28
5 The HDMs for convection-diffusion equations 31
5.1 One dimensional convection-diffusion equation . . . . . . . . 32
5.1.1 The upwind hybrid difference method . . . . . . . . . 33
5.1.2 The penalized upwind method............. 41
5.2 Two dimensional convection-diffusion equation . . . . . . . . 48
5.3 Numerical experiments..................... 55
5.4 Summary ............................ 65
6 The IHD method for elliptic interface problems 66
6.1 One dimensional analysis.................... 66
6.2 Two dimensional IHD method................. 74
6.3 Numerical experiments..................... 84
6.4 Summary ............................ 87
7 Conclusion and future work 88
Appendices 90
A Notations ............................ 90
B The discrete Poincare inequality................ 91
Bibliography 94
