Characteristics method for the formulation and computation of a free boundary cavitation problem

In this paper, a semidiscretized scheme based on characteristics method is analyzed when it is applied to a free boundary problem. The free boundary problem issued from a cavitation model in lubrication involves specific boundary conditions. The idea is to associate to the departure problem a sequence of variational inequalities depending on a discretization parameter. The theoretical convergence result from the solution of the semidiscretized problem to the solution of the continuous one is stated for a flux-imposed boundary condition. For this purpose, due to the specific boundary conditions, technical modifications of the classical version of characteristics are needed. Then, obstacle problem tools are applied to a set of variational inequalities with Neumann boundary conditions in order to obtain L-infinity(Omega) estimates which provide the convergence result. In this way, a result of existence of solution for the continuous problem and a theoretical justification for the usage of the proposed numerical methods are concluded. Finally, some numerical test examples are presented to illustrate the good performance of the method. (C) 1998 Elsevier Science B.V. All rights reserved. AMS classification: 65M; 35Q.