We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While the existing theoretical estimates for the convergence rate of additive Schwarz methods for the $p$-Laplacian are sublinear, the actual convergence rate observed by numerical experiments is linear. In this paper, we bridge the gap between these theoretical and numerical results by analyzing the linear convergence rate of additive Schwarz methods for the $p$-Laplacian. In order to estimate the linear convergence rate of the methods, we present two essential components. Firstly, we present a new 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.

翻译：我们考虑涉及 $p$-Laplacian 的边值问题的加性 Schwarz 方法。尽管现有关于 $p$-Laplacian 加性 Schwarz 方法收敛率的理论估计是次线性的，但数值实验观察到的实际收敛率是线性的。在本文中，我们通过分析 $p$-Laplacian 加性 Schwarz 方法的线性收敛率，弥合了理论与数值结果之间的差距。为了估计该方法的线性收敛率，我们提出了两个关键组成部分。首先，我们提出了一种新的、以拟范数形式表达的加性 Schwarz 方法抽象收敛理论。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 Poincaré–Friedrichs 不等式的拟范数版本，这对于推导双层区域分解设置下的拟范数稳定分解至关重要。