A wavelet-based adaptive mesh refinement method for the obstacle problem

In this paper, a fast computational technique based on adaptive mesh generation for numerical solution of the obstacle problem is considered. The obstacle problem is an elliptic variational inequality problem, where its solution divides the domain into the contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is priori unknown and the solution is not smooth on it. Due to lack of smoothness, numerical methods need a lot of mesh points in discretization for obtaining a numerical solution with a reasonable accuracy. In this paper, using an interpolating wavelet system and the fast wavelet transform, a multi-level algorithm for generating an appropriate-adapted mesh is presented. In each step of the algorithm, the semi-smooth Newton's method or active set method is used for solving the discretized obstacle problem. We test the performance and accuracy of the proposed method by means of four numerical experiments. We show that the presented method significantly reduces CPU time in comparison with the full-grid algorithm and also can simultaneously capture the priori unknown free boundary.