Additive Schwarz Methods for Fourth-Order Variational Inequalities

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. 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 \(H/\delta \) $H/\delta$ only, where h and H are the typical diameters of an element and a subdomain, respectively, and \(\delta \) $\delta$ measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.