A numerical approach for solving a semilinear obstacle problem on the boundary

This paper presents an iterative method for solving a semilinear obstacle problem on the boundary. The approach begins by approximating the problem using a penalty method, transforming it from a problem of the first kind to a sequence of problems of the second kind. The convergence of these approximations is then proven. The approximate problem is reformulated as a set-valued problem and expressed in a mixed formulation, leveraging the sub-differential of a continuous convex function to characterize the contact domain with the obstacle. Finally, we show that both the approximate solution and sub-differential can be effectively computed using an appropriate projection/fixed-point algorithm.