The Numerical Method For A Kind Of Evolutionary Variational Inequality Of Order 4

Obstacle problem and the dynamic obstacle problem are applied widely in physics, mechanics and enginering. The mathematical models of these two problems are formulated as an elliptic variational inequality of order 4 and an evolutionary variational inequality of order 4 with obstacle constraint respectively. The numerical methods for this kind of elliptic variational inequality mainly focus on FEM, duality method, Uzawa method and Domain Decomposition Method. Ordinary we often use time splitting scheme to change the evolutionary variational inequality into an elliptic variational inequality, for instance forword time difference scheme or Newmark method etc. In this paper, two kinds of numerical methods for variational inequality and evolutionary variational inequality are discussed. This thesis is composed of the following sections:In the chapter two, using presents two-grids projection algorithm, two kinds of method for the elliptic problem of variational inequality of order 4. The problem is reduced into an equivalent non-linear equation by the penalty method. Marchuk-Yanenko scheme is applied in the first method. In the second method we directly use Newton method to solve the non-linear penalty equation. Two numerical examples are implemented, The results show the efficiency of the method. It analyzes the effect of the obstacle region, penalty factor to the solutions.In the chapter three, two kinds of methods for the evolutionary variational inequality of order 4 are obtained. The flow charts are presented in this chapter. Two dimensional numerical examples prove the efficiency of this method. We also discuss the effect of obstacle region, parameters to the solution of evolutionary variational inequality. In additional, we compared the efficiency of these two methods.