A Self-Adaptive Uzawa Block Relaxation Algorithm For Solving Unilateral Obstacle Problem

The obstacle problem is also called the free boundary problem.Such problems are described by complementary forms of differential equations and inequalities.It is very difficult to deal with mathematical modeling,theoretical analysis,and numerical computation because of the inherent nonlinearity.In this paper,the Uzawa block relaxation method is extended to solve the unilateral obstacle problem,and self-adaptive rules are proposed to choose the parameter automatically.The main content of this article is organized as follows:The first part is considered with a self-adaptive Uzawa block relaxation method in infinite-dimensional spaces for unilateral obstacle problems.This work introduces an auxiliary unknown and augmented Lagrange function to transform the problem into a saddle-point problem of function spaces which can be solved by the Uzawa block relaxation method.Each iterative step consists of a linear problem while the auxiliary unknown computed explicitly.The convergence speed of this method depends on the penalty parameters heavily,and it is difficult to choose a suitable penalty parameter for individual problems.In order to improve the efficiency of the method,a self-adaptive rule is provided to adjust the penalty parameter automatically.Then a self-adaptive Uzawa block relaxation in infinite-dimensional spaces is obtained for solving unilateral obstacle problems.The convergence of the method is analyzed using elliptic differential operators.The reliability of the algorithm is showed by some numerical results of the propose method.The second part is devoted to a self-adaptive Uzawa block relaxation method in finite-dimensional spaces for unilateral obstacle problems.The obstacle problem is discretized into a finite-dimensional linear complementarity problem,which is equivalent to a saddle-point problem by a finite-dimensional auxiliary variable and an augmented Lagrange function.Then the Uzawa block relaxation method is applied for the solution,and a two-step iterative method is obtained by solving a linear problem as a main subproblem while the auxiliary variable is computed explicitly.The convergence of the method is proved by the matrix positive definiteness.The convergence speed of the method depends significantly on the penalty parameter,and it is difficult to select the appropriate penalty parameter for specific problems.A self-adaptive rule is proposed to improve the perfor-mance of the method.Numerical results confirm the theoretical analysis and effectiveness of the method.The last part summarizes the research content of this paper,and prospects for future research.