论文标题
关于$ p $ -laplacian的添加施瓦茨方法的线性收敛
On the linear convergence of additive Schwarz methods for the $p$-Laplacian
论文作者
论文摘要
我们考虑了涉及$ p $ -laplacian的边界价值问题的加法施瓦茨方法。虽然现有的理论估计值表明这些方法的额定性收敛速率，但来自数值实验的经验证据表明线性收敛速率。在本文中，我们通过提出新的收敛分析来缩小这些理论和经验结果之间的差距。首先，我们提出了一种根据准标准编写的添加剂Schwarz方法的新收敛理论。该准标准表现出类似于与问题相关的凸能功能的Bregman距离的行为。其次，我们提供了poincar'{e} - 弗里德里希（Friedrichs Indorality）的准标准版本，该版本在得出两级域分解设置的准标准稳定分解方面起着至关重要的作用。通过利用这些关键元素，我们建立了$ p $ -laplacian的添加剂schwarz方法的渐近线性收敛。
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 asymptotic linear convergence of additive Schwarz methods for the $p$-Laplacian.
