The Numerical Analysis Of A Kind Of The Time Dependent Friction Problem Using Variational Inequalities

Many problems in the engineering practice can be describe by parabolic variational inequalities, such as mechanical, the physical and optimal control problem. So it is important to study the algorithm of parabolic variational inequalities. In recent years, the numerical method of the variational inequalities gets a rapid development, but most of that is study about elliptic variational inequalities. About parabolic variational inequalities, it is difficult to solve directly because the time derivative term and the non-differentiable term. Because a lot of problems in life can be describe by parabolic variational inequalities, the study of the parabolic variational inequalities is very important and significance.To overcome above difficulties, in this paper, the parabolic variational inequalities can be reduced to an elliptic vatiational inequality by using semi-discretization and implicit method in time, then the non-differentiable approximation by numerical integration. So the calculation of the variational inequalities is more convenient. On the base of that we give the relaxation algorithm and dual algorithm.This paper includes five chapters. In chapter 1, the development of variational inequality, finite element method and solving parabolic variational inequality by the finite element method are mainly introduced; and research dynamic of scholar of home and abroad is emphasized.In chapter 2, a complete set of theories are given, such as the theory of the elliptic variational inequalities, finite element method discrete and so on.In chapter 3, it is presented the parabolic variational inequality problem that is equal to time dependent friction problem. Then the error estimate about the time semi-discretization and the full-discretization are presented.In chapter 4, we study the relaxation algorithm of the parabolic variational inequalities and the convergence of the relaxation algorithm. Finally, the given example proves that the algorithm is effectiveness and feasibility.In chapter 5, we give the dual algorithm about the parabolic variational inequalities and the convergence of the dual algorithm; finally, the numerical example shows feasibility of the method.