Immersed Finite Element Methods For Elliptic PDEs With Interfaces And Irregular Domains

In many real world applications, the considered object is often composed of sev-eral different materials separated from each other by interfaces. If partial differential equations (PDEs) are used to model these applications, the parameters in the govern-ing PDEs are discontinuous, and some interface conditions are needed to be satisfied from physical reasoning or mathematical derivation. In this dissertation, we consid-er some immersed finite element methods (IFEMs) using unfitted meshes for these interface problems.The non-conforming IFEM developed in Li et al. (Numer. Math.96:61-98, 2003) for interface problems was extensively studied in the literature. The non-conforming IFEM is very much like the traditional P1 conforming finite element method but with modified basis functions that enforce the natural jump conditions on interface elements. Although the non-conforming IFEM is simple and has optimal accuracy in L2 and H1 norms, it is not fully second order accurate in L∞ norm owing to the discontinuities of the modified basis functions. While the conforming IFEM also developed in Li et al. (Numer. Math.96:61-98,2003) is fully second order accurate, the implementation is more complicated. In the first part of this dissertation, we develop a symmetric and consistent immersed finite element method (SCIFEM) which improves the original methods. The new method maintains the advantages of the non-conforming IFEM by using the same basis functions, but it is symmetric, con-sistent, and more important, it is second order accurate. The idea is to add some cor-rection terms to the weak form to take into account of the discontinuities in the basis functions. We also extend the SCIFEM to interface problems with non-homogeneous jump conditions.In the second part of this dissertation, we propose two augmented IFEMs which are fast iterative methods for interface problems and problems defined on irregular domains. The augmented approach was first proposed in Li (SIAM J. Numer. Anal. 35:230-254,1998). In that paper, the augmented immersed interface method (AIIM) which is a finite difference method was developed. We simply apply the augment-ed approach to the finite element framework to get the first augmented IFEM. In the augmented method, one or several augmented variables are introduced along the in-terface or boundary so that one can get efficient discretizations. The augmented vari-ables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented method relies on an interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares in-terpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. Based on properties of the finite el-ement method, we then develop the second augmented IFEM that does not need the interpolations. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to problems defined on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jumps in coefficients are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numeri-cal results show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.