Some Progress On Immersed Finite Element Methods For Interface Problems

Interface problems which involve partial differential equations having discontin-uous coefficients across certain interfaces are often encountered in fluid dynamics,electromagnetics, and materials science. For more than four decades, there has been increasingly interest in interface problems, and a vast of literature is available. To the present, there are two major classes of finite element methods (FEM) for inter-face problems, namely, interface-fitted FEM and interface-unfitted FEM categorized according to the topological relation between discrete elements and the interface. In this dissertation, we consider some immersed finite element methods (IFEMs) using unfitted meshes for these interface problems.In Chapter 1, we give an overview of the interface problems. we start with the typical second elliptic order equation that appears in many applications. A result of how accurate one could achieve numerically in an interface-fitted mesh for elliptic interface problems is given. Then, we introduce some classic methods using interface-unfitted mesh for solving interface problems. We give the introduction of the Sobolev space at the end.In Chapter 2, we propose super-convergence finite element methods for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side). The key in 1D is to use a simple weak form to get second order accurate fluxes at the interface from each side. For 2D interface problems, the idea is to intro-duce a small tube near the interface and introduce the gradient as part of unknowns,which is similar to a mixed finite element method, except only at the interface. Thus the computational cost is just slightly higher than the standard finite element method.We present rigorous one dimensional analysis, which show second order convergence rates for both of the solution and the gradient in 1D. For two dimensional problems,we present numerical results and observe second order convergence for the solution,and super-convergence for the gradient at the interface.In Chapter 3, we present a new nonconforming immersed finite element (IFE)method on triangular Cartesian meshes for solving planar elasticity interface prob-lems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover,the method is robust with respect to the shape of the interface and its location rela-tive to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimental-ly, extensive numerical examples are given to show that the convergence orders in L2 norm and semi-H1 norm are optimal under various Lame parameters settings and different interface geometry configurations.In Chapter 4, we develop a new finite element method for solving boundary val-ue problems of 4th order partial differential equations with discontinuous coefficients with interface-unfitted mesh. In the method, the standard basis functions of the Mor-ley element are used for non-interface elements and piecewise basis functions are constructed according to the location of the interface and pertinent jump conditions for interface elements. We investigate some basic properties of the proposed method.Numerical experiments are presented to show that the optimal approximation capabil-ity and the IFE solution also converges optimally.In Chapter 5, a fast finite difference method is developed for solving 4th order partial differential equations with discontinuous coefficients. The method is based on an augmented approach by introducing an intermediate (augmented variable) bound-ary condition along the boundary so that the problem can be treated as two separated Poisson equations with jumps in the source terms along the interface. Thus a fast Poisson solver can be utilized, which makes the proposed method fast. Numerical experiments against analytic solutions show that the computed solution using the pro-posed method has second order accuracy (convergence) in the maximum norm.