Iteration–discretization methods for variational inequalities over fixed point sets

In the last few years a wide range of iterative methods has been developed to treat variational inequalities over fixed point sets in Hilbert spaces. As a rule computational handling of problems in infinite dimensional Hilbert spaces in addition requires some discretization. Any useful discretization of the original leads to families of variational inequalities over families of fixed point sets over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration, are embedded into each other. In the present paper this task is addressed by an iterative method with only a finite number of steps of the proposed iterative method in each of the discrete spaces. From the algorithmic point of view these methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. As an illustration we later apply the method to a simple control problem with elliptic state equations and some bound on the controls. As the discretization technique for the state equation a nested family of piecewise linear C0C0-elements conforming finite element discretizations is used. These discretizations provide inner approximations of the underlying fixed point problems for the variational inequalities. The abstract convergence analysis given in the first part covers the considered illustrative example.