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类科学与工程学科中，包括椭圆最优控制问题和板障碍问题。本文研究求解四阶变分不等式的加性Schwarz方法。基于处理四阶变分不等式的多种有限元方法的统一框架，我们发展了一级和两级加性Schwarz方法。我们证明了两级方法具有可扩展性，即该方法的收敛速度仅依赖于$H/h$和$H/\delta$，其中$h$和$H$分别为单元和子域的典型直径，$\delta$衡量子域间的重叠程度。该证明依赖于一个全新的非线性保正值粗网格插值算子，其构造此前尚属未知。据我们所知，该分析是首次对求解四阶变分不等式的两级加性Schwarz方法可扩展性的研究。数值实验验证了我们的理论结果。