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：四阶变分不等式广泛出现于科学与工程领域，包括椭圆最优控制问题与板障碍问题。本文研究用于求解四阶变分不等式的加性施瓦茨方法。基于四阶变分不等式各类有限元方法的统一框架，我们发展了一水平与两水平加性施瓦茨方法。我们证明了两水平方法具有可扩展性，其收敛速率仅依赖于 $H/h$ 与 $H/\delta$，其中 $h$ 与 $H$ 分别表示单元与子区域的典型直径，$\delta$ 度量子区域间的重叠程度。该证明依赖于一种新型的非线性保正粗插值算子，其构造方法此前尚未知晓。据我们所知，此分析首次研究了两水平加性施瓦茨方法在四阶变分不等式中的可扩展性。数值实验验证了我们的理论结果。