On the convergence of a multigrid method for Moreau-regularized variational inequalities of the second kind

We analyze the behavior of a multigrid algorithm for variational inequalities of the second kind with a Moreau-regularized nondifferentiable term. First, we prove a theorem summarizing the properties of the Moreau regularization of a convex, proper, and lower semicontinuous functional that is used in the rest of the paper. We prove that the solution of the regularized problem converges to the solution of the initial problem when the regularization parameter approaches zero. To give a procedure of explicit writing of the Moreau regularization of a convex and lower semicontinuous functional, we have constructed the Moreau regularization for two problems with a scalar unknown taken from the literature and also, for a contact problem with Tresca friction. These functionals are of an integral form and we prove some propositions giving general conditions for which the functionals of this type are lower semicontinuous, proper, and convex. To solve the regularized problem, which is a variational inequality of the first kind, we use a standard multigrid method for two-sided obstacle problems. The numerical experiments have showed a high accuracy and a very good convergence of the method even for values of the regularization parameter close to zero. In view of these results, we think that the proposed method can be an alternative to the existing multigrid methods for the variational inequalities of the second kind.