声明
Contents
摘要
Abstract
Introduction
Chapter 1 Preliminaries
1．1 Model problems and applications
1．2 The level set representation of interfaces
1．3 Unfitted meshes
1．4 Error analysis of the Pa conforming finite element method using unfitted meshes for interface problems
1．5 A numerical example
Chapter 2 The SCIFEM for the elliptic interface problem with homogeneous jump conditions
2．1 A brief review of the IFE space and related IFE methods
2．2 The symmetric and consistent IFE method and error analysis
2．3 Numerical experiments
Chapter 3 The SCIFEM for the elliptic interface problem with non-homogeneous jump conditions
3．1 Homogeneous and non-homogeneous IFE spaces
3．2 The non-conforming IFE method
3．3 The Petrov-Galerkin IFE method
3．4 The symmetric and consistent IFE method
3．5 Numerical experiments
Chapter 4 Augmented Immersed Finite Element Methods
4．1 A brief review of the original AIIM
4．2 The singularity removal finite element method for Poisson equations with singular sources
4．2．1 Some implementation details
4．2．2 The convergence proof in the broken H1-norm
4．3 Numerical experiments
Chapter 5 A New Augmented Immersed Finite Element Method Without Using SVD Interpolations
5．1 The new augmented IFEM for elliptic interface problems defined on a rectangular domain
5．1．1 The augmented technique in the continuous case
5．1．2 The augmented technique in the discrete case
5．2 Generalizations of the new method
5．2．1 The new SVD-free augmented IFEM for Poisson equations on irregular domains with a Dirichlet boundary condition
5．2．2 The new SVD-free augmented IFEM for general interface problems on irregular domains
5．3 Numerical experiments
5．3．1 Numerical experiments for interface problems on rectangulardomains
5．3．2 A numerical example of Poisson equation on an irregular domain
5．3．3 A numerical example of an interface problem on an irregular domain
Bibliography
Publications and Finished Papers
Acknowledgements
