2022/10/17 by Young-Ju Lee, Lee, Young-Ju, Jong-Ho Park +1 · 2 citations
Computer Science · Engineering · Mathematics · #65J15 #65K15 #65N55 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2210.09183
openalex publication_date 2022/10/17 · openalex created_date 2022/10/20 · openalex updated_date 2026/07/28
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.