Finite Element-Finite Difference Methods For Several Anisotropic Interface Problems

Many physical phenomena can be modeled as anisotropic interface problems of partial differential equation and can be numerically solved and simulated.Examples include reservoir modeling and un-derground water flow with anisotropic permeabilities,option pricing problems that often involve free boundary with cross derivatives(anisotropic)and infinite domains,crystal growth and Hele-Shaw flows,Stefan problems,etc.For these problems,the coefficients representing the properties of different media are discontinuous,and the solution and its derivatives may be non-smooth or even discontinuous.Thus,it is important and challenging to find accurate numerical solutions for anisotropic interface problems.If a standard finite element method is utilized,it is difficult to guarantee the accuracy of the numerical solu-tion near or on an interface.If a standard finite difference method is applied,it is challenging to prove the stability,so its convergence,of the method due to mixed derivative terms.In this thesis,we develop some hybrid finite element-finite difference methods based on Cartesian meshes to solve anisotropic elliptic and parabolic interface problems.In the Chapter 1,we give a brief introduction on the background and significance of anisotropic interface problems,followed by an overview of numerical methods developed to these problems.The governing equations of anisotropic elliptic and parabolic interface problems are also introduced,which are solved by some immersed interface methods(IIM).Therefore,the basic idea and implementation process of IIM are described briefly.Some proposed methods of this thesis are showed at the end.In chapter 2,a new finite element-finite difference(FE-FD)method is developed for two-dimensional anisotropic elliptic interface problems.The idea is to applied finite element discretization away from the interface so that the coefficient matrix is symmetric positive(or semi-positive);and a finite difference maximum principle preserving method on interface triangles so that the part of coefficient matrix is an M-matrix.Error analysis is carried out based on the theory of finite element methods and the comparison theorem of finite difference methods.An interpolation scheme based on the immersed interface method is also applied to compute the normal derivative of solution(or gradient)accurately from each side of the interface.Numerical experiments are also presented.In chapter 3,a second order accurate method in the infinity norm is proposed for general three di-mensional anisotropic elliptic interface problems in which the solution and its derivatives,the coefficients,and source terms all can have finite jumps across one or several arbitrary smooth interfaces.The method is based on the 2D finite element-finite difference(FE-FD)method but with substantial differences in method derivation,implementation,and convergence analysis.One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differ-ential equations and the jump conditions.A finite element discretization whose coefficient matrix is a symmetric positive definite is used away from the interface;and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through.We aim to get a sharp interface method that can have second order accuracy in the point-wise norm.We show the convergence analysis by splitting errors into several parts.Nontrivial numerical examples are presented to confirm the convergence analysis.In chapter 4,a new sharp interface method is proposed for anisotropic parabolic partial differential equations with moving interfaces on Cartesian meshes.In the spatial discretization,the second order finite element-finite difference(FE-FD)strategy is applied that guarantees the resulting matrix is symmetric positive definite at all regular grid points while the other part is a M-matrix for all irregular grid points.The stability and accuracy is further enforced when a modified Crank-Nicolson discretization is applied in the time discretization.Numerical experiments are presented to show second order convergence of the computed solution.In chapter 5,an augmented finite element-finite difference method is proposed for anisotropic elliptic interface problems.The main idea is to extend the augmented immersed interface method for anisotropic interface problems.Using two augmented variables(the jumps of first order and second order normal derivatives on the interface),the original problem is reduced to a system of three equations.For the first governing equation,a finite element-finite difference scheme developed in chapter 2 is used with a correction in right hand side of the discrete linear system of equations,in which the correction term is dependent on the splitting form of the jump conditions in two axes,and it is established from correcting the difference scheme along three different directions.The other two augmented equations defined only on the interface are discretized by a IIM based interpolation approach and solved by a GMRES method.Numerical experiments are presented to show the effectiveness of the new method.In chapter 6,two simple third and fourth order compact finite element methods are proposed for one-dimensional Sturm-Liouville boundary value problems.The key idea is based on the interpolation error estimate,which can be related to the source term.Thus,a simple posterior error analysis or modified basis functions based on original piecewise linear or quadratic basis function will lead to a higher order accurate solution in the L2 norm,and in the Hl or the energy norm.Numerical examples have confirmed our analysis.