Domain Decomposition Methods For A Kind Of Variational Inequalities

Variational inequalities arise from mathematical modelling of physical, mechanical and economical applications such as fluid flow in porous media, the behavior of the elasto-plastic materials, lubrification phenomena in mechanics, friction problems, and the spatial problems resulting from the implicit time-discretization of two phase Stefan problems, etc. With the development of science and computing technique, many algorithms for solving the variational inequalities have been constructed and analyzed. Especially with the development of parallel computing technique, corresponding numerical and theoretical researches have made great progress.Domain decomposition methods, which have been widely studied recently, have the advantages of reducing the large-scale problem into smaller ones, reducing the complex problems into simpler ones and turning serial computing into parallel computing. With the development of parallel technique, the study of these methods has been greatly stimulated.The theories concerned with the variational inequlities of the first kind have been established systematically. On the other hand, as for the variational inequalities of the second kind , both the algorithms construction and the theoretical analysis are needed. Researches in this fields are relatively few.In this paper, we firstly give a review of the finite element approximation and error estimate of variational inequalities of the second kind.Secondly, we propose multiplicative and additive domain decomposition methods for the solution of variational inequalities of the second kind. The convergence of the methods are obtained.Lastly, numerical examples based on the algorithms presented in this paper are given and the results show these algorithms are effective.