| 摘要 | 第8-10页 |
| Abstract | 第10-11页 |
| Introduction | 第12-16页 |
| Chapter 1 Preliminaries | 第16-29页 |
| 1.1 Model problems and applications | 第16-19页 |
| 1.2 The level set representation of interfaces | 第19-20页 |
| 1.3 Unfitted meshes | 第20-22页 |
| 1.4 Error analysis of the P_1 conforming finite element method using unfitted meshes for interface problems | 第22-27页 |
| 1.5 A numerical example | 第27-29页 |
| Chapter 2 The SCIFEM for the elliptic interface problem with homoge-neous jump conditions | 第29-56页 |
| 2.1 A brief review of the IFE space and related IFE methods | 第29-33页 |
| 2.2 The symmetric and consistent IFE method and error analysis | 第33-43页 |
| 2.3 Numerical experiments | 第43-56页 |
| Chapter 3 The SCIFEM for the elliptic interface problem with non-homogeneous jump conditions | 第56-74页 |
| 3.1 Homogeneous and non-homogeneous IFE spaces | 第57-60页 |
| 3.2 The non-conforming IFE method | 第60-61页 |
| 3.3 The Petrov-Galerkin IFE method | 第61页 |
| 3.4 The symmetric and consistent IFE method | 第61-63页 |
| 3.5 Numerical experiments | 第63-74页 |
| Chapter 4 Augmented Immersed Finite Element Methods | 第74-92页 |
| 4.1 A brief review of the original AIIM | 第74-78页 |
| 4.2 The singularity removal finite element method for Poisson equations with singular sources | 第78-85页 |
| 4.2.1 Some implementation details | 第81-83页 |
| 4.2.2 The convergence proof in the broken H~1-norm | 第83-85页 |
| 4.3 Numerical experiments | 第85-92页 |
| Chapter 5 A New Augmented Immersed Finite Element Method Without Using SVD Interpolations | 第92-114页 |
| 5.1 The new augmented IFEM for elliptic interface problems defined on a rectangular domain | 第92-100页 |
| 5.1.1 The augmented technique in the continuous case | 第92-94页 |
| 5.1.2 The augmented technique in the discrete case | 第94-100页 |
| 5.2 Generalizations of the new method | 第100-102页 |
| 5.2.1 The new SVD-free augmented IFEM for Poisson equations on irreg-ular domains with a Dirichlet boundary condition | 第100-101页 |
| 5.2.2 The new SVD-free augmented IFEM for general interface problems on irregular domains | 第101-102页 |
| 5.3 Numerical experiments | 第102-114页 |
| 5.3.1 Numerical experiments for interface problems on rectangular do-mains | 第103-111页 |
| 5.3.2 A numerical example of Poisson equation on an irregular domain | 第111-112页 |
| 5.3.3 A numerical example of an interface problem on an irregular domain | 第112-114页 |
| Bibliography | 第114-119页 |
| Publications and Finished Papers | 第119-120页 |
| Acknowledgements | 第120页 |
