Overlapping doman decomposition methods for the obstacle problem

In this paper overlapping domain decomposition methods are applied to the numerical solution of nonlinear grid variational problems arising from the approximation of the obstacle problem by the piecewise linear finite element method. This method is important for nonlinear boundary value problems for two reasons: It provides the possibility of using parallel processing, and, what is perhaps more important, the means for isolating the neighbourhood of the free boundary for a special treatment. In the major part of the domain the problem is linear and traditional efficient solvers for linear problems can be applied. We give the sufficient conditions of the convergence of the method and formulate the convergence result. Moreover, we give some considerations about overlapping domain decomposition methods with monotone operations.