We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. Although the existing theoretical estimates indicate a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper, we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. Firstly, we present an abstract convergence theory of additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior similar to the Bregman distance of the convex energy functional associated to the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which is essential for deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these two key elements, we establish a new bound for the linear convergence rate of the methods.

翻译：我们考虑涉及 $p$-Laplacian 的边值问题的加法型 Schwarz 方法。尽管现有的理论估计表明这些方法具有次线性收敛速率，但数值实验的经验证据显示其具有线性收敛速率。在本文中，我们通过提出一种新的收敛性分析来缩小理论与经验结果之间的差距。首先，我们提出了一种基于拟范数描述的加法型 Schwarz 方法的抽象收敛理论。该拟范数的行为类似于与问题相关的凸能量泛函的 Bregman 距离。其次，我们给出了拟范数形式的 Poincaré--Friedrichs 不等式，这对于推导两层区域分解设置下的拟范数稳定分解至关重要。通过利用这两个关键要素，我们建立了方法线性收敛速率的新上界。