Two-level methods with optimal computing complexity for variational inequalities of the second kind

We introduce and analyze some two-level multiplicative and additive Schwarz methods for variational and quasi-variational inequalities of the second kind. The methods are introduced as subspace correction algorithms for problems in a reflexive Banach space. We prove that these methods are globally convergent and give, under some assumptions, error estimates. In the finite element spaces, the introduced algorithms are in fact two-level Schwarz methods. In this case we prove that the assumptions we made for the general convergence result hold, and write the convergence rate depending on the overlapping and mesh parameters. We get that our methods have an optimal convergence rate, it is almost independent of the mesh and overlapping parameters, and also, the methods have an optimal computing complexity per iteration.