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 $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ 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.

翻译：四阶变分不等式出现在各种科学与工程学科中，包括椭圆最优控制问题和板障碍问题。本文研究了求解四阶变分不等式的加性Schwarz方法。基于四阶变分不等式的多种有限元方法的统一框架，我们发展了单层和双层加性Schwarz方法。我们证明了双层方法具有可扩展性，即该方法的收敛速度仅依赖于$H/h$和$H/\delta$，其中$h$和$H$分别表示单元和子域的典型直径，$\delta$衡量子域间的重叠程度。该证明依赖于一个新的非线性保正粗插值算子，其构造此前尚属未知。据我们所知，该分析首次探讨了双层加性Schwarz方法用于四阶变分不等式的可扩展性。数值实验验证了我们的理论结果。