On the linear convergence of additive Schwarz methods for the $p$-Laplacian

Abstract

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.

26 pages, 8 figures

References in corpus (1)

• Additive Schwarz Methods for Convex Optimization with Backtracking

Cited by in corpus (3)

• Additive Schwarz methods for fourth-order variational inequalities
• Two-level overlapping Schwarz preconditioners with universal coarse spaces for $2m$th-order elliptic problems
• Parallel subspace correction methods for semicoercive and nearly semicoercive convex optimization with applications to nonlinear PDEs