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  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 , using, among other things, ideas arising from splitting methods (see Marchuk  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  for equations of evolution. We show here how it can be adapted to variational inequalities (what is needed from  is recalled here).
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Pierre-Louis Lions, Parallel algorithms for the solution of variational inequalities. Interfaces Free Bound. 1 (1999), no. 1, pp. 3–16DOI 10.4171/IFB/1