Additive Schwarz methods for fourth-order variational inequalities

We consider additive Schwarz methods for fourth-order variational inequalities. While most existing works on Schwarz methods for fourth-order variational inequalities deal with auxiliary linear problems instead of the original ones, we deal with the original ones directly by using a nonlinear subspace correction framework for convex optimization. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. To the best of our knowledge, the proposed two-level method is the first scalable Schwarz method for fourth-order variational inequalities. An efficient numerical method to solve coarse problems in the two-level method is also presented. Our theoretical results are verified by numerical experiments.