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:我们考虑在$p$-拉普拉斯方程边值问题中的加性Schwarz方法。尽管现有的关于$p$-拉普拉斯方程加性Schwarz方法收敛速度的理论估计是次线性的,在数值实验中观察到的实际收敛速度是线性的。本文通过分析$p$-拉普拉斯方程中加性Schwarz方法的线性收敛速度,弥合了这些理论和数值结果之间的差距。为了估计方法的线性收敛速度,我们提出了两个基本组成部分。首先,我们提出了一种新的加性Schwarz方法的抽象收敛理论,用拟范数来表示。这个拟范数表现出类似于与问题相关的凸能量泛函的Bregman距离的行为。其次,我们提供了一种拟范数版的Poincar'{e}--Friedrichs不等式,这是推导两级域分解设置的拟范数稳定分解的关键。