Parallel algorithms for the solution of variational inequalities

Abstract

One of the many ways of solving free-boundary problems is, when possible, to put them (perhaps after suitable transformations) in the framework of variational or quasi-variational inequalities. It then remains to solve them numerically, a task which has been studied by Glowinski, Lions, & Tremolieres [9] without reference to parallel algorithms. On the other hand, systematic attempts to decompose the problems of the calculus of variations and of control theory have been made by Bensoussan, Lions, & Temam [4], using, among other things, ideas arising from splitting methods (see Marchuk [25] and the bibliography therein). We propose here a general method for obtaining, in infinitely many ways, stable parallel algorithms for the solution of variational inequalities of evolution. This method was introduced in [12] for equations of evolution. We show here how it can be adapted to variational inequalities (what is needed from [12] is recalled here).