We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest 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 a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements, we establish the linear convergence.

翻译：我们考虑涉及$p$-拉普拉斯算子的边值问题的加性Schwarz方法。现有理论估计表明这些方法具有次线性收敛率，但数值实验的经验证据却显示出线性收敛率。本文通过提出一种新的收敛性分析，缩小了理论与实证结果之间的差距。首先，我们提出了一种基于拟范数的加性Schwarz方法新收敛理论。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，我们给出了拟范数版本的Poincaré-Friedrichs不等式，该不等式在推导两层区域分解设定的拟范数稳定分解中起着关键作用。通过利用这些关键要素，我们建立了线性收敛性。