| 摘要 | 第7-9页 |
| Abstract | 第9-10页 |
| Chapter 1 Introduction | 第11-21页 |
| 1.1 An overview of interface problems | 第11-14页 |
| 1.2 Model problems and applications | 第14-16页 |
| 1.3 Interface-fitted method | 第16-17页 |
| 1.4 Interface-unfitted method | 第17-18页 |
| 1.5 Sobolev Space | 第18-21页 |
| Chapter 2 Accurate gradient computations at interfaces using finite ele-ment methods | 第21-43页 |
| 2.1 Model problems and applications | 第21-24页 |
| 2.2 One-dimensional algorithm and analysis | 第24-36页 |
| 2.2.1 The immersed finite element method in 1D | 第25-26页 |
| 2.2.2 Convergence analysis of 1D IFEM | 第26-31页 |
| 2.2.3 An accurate flux computation at the left of the interface | 第31-35页 |
| 2.2.4 Numerical experiments in 1D | 第35-36页 |
| 2.3 The numerical method and experiments for the 2D interface problem | 第36-43页 |
| 2.3.1 Numerical experiments in 2D | 第38-43页 |
| Chapter 3 A Cartesian grid nonconforming immersed finite element methodfor planar elasticity interface problems | 第43-76页 |
| 3.1 Model problems and applications | 第43-48页 |
| 3.2 A nonconforming P_1/CR IFE method | 第48-56页 |
| 3.2.1 The nonconforming P_1/CR IFE basis functions on non-interface el-ements | 第49-50页 |
| 3.2.2 The nonconforming P_1/CR IFE basis functions on interface ele-ments | 第50-55页 |
| 3.2.3 The immersed nonconforming P_1/CR Galerkin method and the IFEinterpolation | 第55-56页 |
| 3.3 Properties of the nonconforming P_1/CR IFE space | 第56-60页 |
| 3.4 Numerical experiments | 第60-76页 |
| Chapter 4 An unfitted immersed Morley element method for fourth orderelliptic equations | 第76-91页 |
| 4.1 The model problems and applications | 第76-78页 |
| 4.2 A Morley IFE method | 第78-85页 |
| 4.2.1 Basis functions of the Morley element on non-interface elements | 第79-80页 |
| 4.2.2 Basis functions of the Morley element on interface elements | 第80-84页 |
| 4.2.3 The immersed Morley Galerkin method and the IFE interpolation | 第84-85页 |
| 4.3 Unisolvent property for the basis functions of the Morley IFE | 第85-87页 |
| 4.4 Numerical experiments | 第87-91页 |
| Chapter 5 An augmented method for 4th order PDEs with discontinuouscoefficients | 第91-105页 |
| 5.1 Introduction | 第91-95页 |
| 5.2 The augmented formulation | 第95-96页 |
| 5.3 The augmented numerical algorithm | 第96-99页 |
| 5.4 Numerical experiments | 第99-105页 |
| Chapter 6 Conclusions and future work | 第105-106页 |
| Bibliography | 第106-112页 |
| Publications and Completed Papers | 第112-113页 |
| Acknowledgements | 第113-114页 |
