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.

翻译：我们考虑了四级变异性不平等的添加物施瓦兹方法。虽然关于四级变异性不平等方法的现有大部分工作涉及辅助线性问题,而不是原有的线性问题,但我们通过使用非线性子空间校正框架直接处理原始问题,对锥形进行优化。根据四级变异性不平等各种有限元素方法的统一框架,我们开发了一级和二级的添加物施瓦兹方法。我们证明,两级方法的趋同率是可伸缩的,因为这种方法的趋同率仅取决于H/h$和H/delta美元,其中,美元是元素和子体的典型直径,而美元则测量子体之间的重叠。据我们所知,拟议的两级方法是第四级变异性不平等的第一个可伸缩的施瓦兹方法。还提出了解决二级方法中粗糙问题的高效数字方法。我们的理论结果通过数字实验得到验证。