Finite Difference Domain Decomposition Algorithm

As a new effective algorithm to slove partial differential equation,domain decomposition algrithm is attracting more and more attentions.The algorithm divides the computational domain into several subdomains,and solves the original problem in the sub-domains.On one hand ,owing to using different discrete methods in different subdomains according to the characters of the submodel,the algorithm will improve the computational accuracy,especially for the subdomains whose solutions change largely;On the other hand,the algorithm can slove differential equation in every subdomain indepen-dently,which improve the computational speed greatly.The key of the algorithm is how to fix on the boundary values of the subdomains and how to put the solutions of every subdomains together,such that it can turn into a rational approximation of the original problem.There are many research achievements with regard to the domain decomposition algorithm,among which the most representative results are D-N algorithm and N-N algorithm based on finite element method, unbounded domain decomposition algorithm based on boundary element method.There are four chapters in this paper.The chapter 1 is introduction,which introduces the basic knowledge of finite difference domain decomposition algorithm briefly and summarizes the basic content of the paper.In chapter 2,we present the finite difference domain decomposition algorithm of two-dimensional changing coefficient elliptical equation.Furthermore in §2.3 we introduce proper bilinear function and norm,which are used to carry out the error estimate of the difference format of two-dimensional changing coefficient elliptical equation so as to gain the error rank O(Δx~2 + Δy~2) .In chapter 3,we extend the differece format in chapter 2 to two-dimensional changing coefficient parabolic equation. We propose the corresponding difference format of two-dimensional changing coefficient parabolic equation ,and carry out the error estimation of the formate.We obtain the error rank O(Δx~2 + Δy~2 + Δt). In chapter 4,we extend the analogous domain decomposition and difference format to two-dimensional changing coefficient hyperbolic equation,and do the corresponding format error estimation,which results in the error rank 0(Δx~2 + Δy~2 + Δt~2).