A Non-overlapping Domain Decomposition Method For A Kind Of Nonlinear Variational Inequalities

The focus of this dissertation is to study domain decomposition methods for a kind of variational inequalities.Variational inequality problems have many applica-tions. Much attention has been drawn to their numerical solutions from numerical analyzers and engineers. In this paper.we study numerical method for a kind of non-linear variational inequalities.These variational inequalities may come from the free boundary problems, in which the solution boundary contains a part of priori unknown boundary, called free boundary.For the corresponding variational inequality of the free boundary problem, we will construct a domain decomposition method and discuss Its convergence.Domain decomposition methods are developed rapidly since80’s of last century. Same as MG. domain decomposition methods have a very good property of mesh in-dependent convergence, and therefore are called "fast solvers’. These methods can be divided into overlapping and non-overlapping domain decomposition methods accord-ing to if there is or no overlapping between the sub-domains. The method we proposed belongs to non-overlapping domain decomposition methods.For the free boundary problem with a nonlinear elliptic partial differential opera-tor.based on its equivalent variational inequality.we will construct a non-overlapping domain decomposition method. The method can be regarded as an extension for the the discussions relating to the free boundary problems with a linear elliptic partial differential operator. The article concludes the following:Firstly in Chapter one. we introduced the research background of free boundary problems and the development of domain decomposition methods. For the free bound-ary problem we considered. we present its equivalent variational inequality form. As a basement, some results for the free boundary problem with a linear elliptic partial differential operator are also mentioned for completion.In Chapter two. we propose a non-overlapping domain decomposition method for the nonlinear problem and establish its convergence theorem.Lastly in Chapter tree, we illustrate some numerical experiments to investigate the effieicney of the methods and verify the convergent theoretical result we obtains. Tn the tests. a Miehaelis-Menten reaction diffusion equation in two dimension with time delav is concluded.