Iterative approximations of penalty operators

Abstract

A well known approach to approximate a variational inequality consists of using a penalty operator (nonlinear in general) [3, 8]. On the other hand, it is sometimes possible to use iterative approaches [1, 4, 9]. In this work an iterative equation with linear penalty operator associated with a variational inequality is constructed. The convergence of the solutions and the error estimates are proved.

Further, primary iterative procedures based on these results are proposed to find approximate solutions of variational inequalities. Estimates of the error and the iteration numbers are obtained.

These investigations were suggested by a study of the contact and plastic problems in solid mechanics [5]. With the described methods, approximate solutions of the contact elastoplastic problems for a plate are obtained [6,7].