Two-Level Schwarz Method For Obstacle Problem

A new two-level additive Schwarz algorithm for obstacle problem is proposed onthe base of Pasi Tarvainen's two-level multiplicative Schwarz algorithm. The additivealgorithm is necessary and important,because unlike the normal domain decompositionmethods for partial di?erential equations,the domain partitions in the two-level Schwarzalgorithm are dependent on the free boundary of the problem.That is to say,we can'tdecompose the subdomain ?exibly. In this paper,the convergence theory of the two-leveladditive Schwarz algorithm is established,moreover,the two-level Schwarz algorithmfor obstacle problem with a nonlinear source term is constructed,and the correspondingconvergence theorem is established.This thesis has four parts.In the first chapter,the background and the necessity for the study of domain de-composition are presented.Then,some known results of two-level Schwarz method areintroduced.And the main results of this paper are also simply introduced.In the second chapter,the whole two-level additive Schwarz algorithm and its con-vergence results are given.In the third chapter,the two-level Schwarz algorithm for the obstacle problem witha nonlinear source term is presented and the corresponding convergence of the algorithmis given.The last chapter is the numerical experiments,in this chapter,some numerical re-sults are given to test the new algorithm.Comparison to some existing Schwarz domaindecomposition method is also given.There are two innovation in this thesis:First,Pasi Tarvainen's idea is extended frommultiplicative algorithm to the additive algorithm;Second,the algorithm for linear com-plementary problem is generalized to the obstacle problem with a nonlinear term.