Numer. Math. 51,631-654 (1987) Numerische Ma emaUk 9 Springer-Verlag 1987 Applications of Fixed-Point Methods to Discrete Variational and Quasi-Variational Inequalities* S. A. Belbas 1 and I. D. Mayergoyz 2 Department of Mathematics, University of Alabama, University, AL 35486, USA 2 Electrical Engineering Department, and The Institute for Advanced Computer Studies, Universityof Maryland, College Park, MD 20742, USA Summary. In this paper, discrete analogues of variational inequalities (V.I.) and quasi-variational inequalities (Q.V.I.), encountered in stochastic control and mathematical physics, are discussed. It is shown that those discrete V.I.'s and Q.V.I.'s can be written in the fixed point form x = Tx such that either T or some power of T is a contraction. This leads to globally convergent iterative methods for the solution of discrete V.I.'s and Q.V.I.'s, which are very suitable for implementation on parallel computers with single-instruction, multiple-data architecture, particularly on massively parallel processors (M.P.P.'s). Subject Classifications: AMS(MOS) : 65N20, 65N10; CR: G1.8. I. Introduction In this paper, we are concerned with the iterative solution of discrete variational and quasi-variational inequalities. These problems are obtained, via finite difference discretization, from continuous variational and quasi-variational inequalities. However, the discrete problems are sometimes interesting in their own right, since they can arise in some applications independently of continuous problems. We present below a brief description of the problems we consider and we will describe first the continuous variational and quasi-variational inequalities in the so-called "pointwise" form, which can be justified once sufficient regularity of the solutions has been established. Variational and quasi-variational inequalities arise in optimal stochastic control [7, 8], as well as in other problems in mathematical physics, e.g. deformation of elastic bodies stretched over solid obstacles, elasto-plastic torsion, etc. [15]. This research is in part supported by the U.S. Department of Energy, Engineering Research Program, under Contract No. DE-AS05-84EH13145