Numerical Algorithms 33: 241–250, 2003.  2003 Kluwer Academic Publishers. Printed in the Netherlands. Numerical simulation of the two-hydrodynamic film thickness M. El Alaoui Talibi ∗ and A. El Kacimi Department of Mathematics, Numerical Analysis and Optimization Laboratory, Université Cadi Ayyad, F.S.S.M., B.P. 2390, Marrakesh, Maroc E-mail: {elalaoui;akacimi}@ucam.ac.ma Received 7 January 2002; accepted 28 October 2002 We investigate, from a numerical point of view, the coefficients identification problem, for the Elrod–Adams model of cavitation, in the frame of the hydrodynamic lubrication. We relax the control problem, and propose a relaxed augmented Lagrangian algorithm, with a given loop length. Some numerical results are given. Keywords: optimal control, coefficients identification, lubrication, nonlinear elliptic problem, relaxation, relaxed augmented Lagrangian algorithm AMS subject classification: 49J40, 76M30, 35R35, 76D08 1. Introduction This work deals with the numerical solution of a coefficients identification prob- lem, in the frame of the hydrodynamic lubrication. The cavitation phenomenon is de- scribed by the Elrod–Adams model [14]. This leads to a free boundary problem that we cannot model by a variational inequality. Our objective is to rebuild the film thickness of a lubricated device, from a given pressure distribution and lubricant concentration. Sim- ilarly to the variational inequality case [7–9,18], we interpret the ‘multi-state’ equation as a complementarity problem, where the lubricant concentration is considered as a sup- plementary control variable. We adopt the relaxation of constraints approach mentioned in [6–8]. From a numerical point of view [6–8], this technique allows to obtain optimal- ity conditions which are easy to exploit in the numerical experiments. Theoretically, it is introduced to ensure a constraints qualification condition [13]. We define the Lagrangian associated to the relaxed problem, to which we add some penalization terms of the state equation and of the relaxed mixed non convex inequality constraint. Then we propose an algorithm of augmented relaxed Lagrangian, with a given a loop length [6,16], per- mitting to resolve the primal problem sufficiently. Some numerical results are given to make valid the performance of the method. ∗ Corresponding author.