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

TL;DR

The paper tackles the nonlinear $p$-Laplacian and the observed linear convergence of additive Schwarz methods despite existing sublinear theoretical rates. It introduces a problem-adapted quasi-norm $\|\cdot\|_{(\nabla v)}$ that closely approximates the Bregman distance $D_F(u,v)$ via $\mu_p \|u-v\|_{(\nabla v)}^2 \le D_F(u,v) \le L_p \|u-v\|_{(\nabla v)}^2$, and develops a quasi-norm Poincaré--Friedrichs inequality to enable a quasi-norm stable decomposition for a two-level domain decomposition. Using a generalized additive Schwarz lemma and a careful control of the quasi-norm terms, the authors prove asymptotic linear convergence of the two-level additive Schwarz method for the $p$-Laplacian, under the condition that the gradient of the exact solution does not vanish. Numerical experiments corroborate the linear convergence in practice and demonstrate scalability with respect to the subdomain overlap, aligning theory with observed performance. Together, these results provide a foundational, problem-tailored convergence theory for nonlinear subspace correction methods and suggest avenues for acceleration and extension to broader nonlinear systems.

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.

Figures (8)

• Figure 1: Discretization and domain decomposition settings when $h = 1/2^4$, $H = 1/2^2$, and $\delta = h$. (a) Coarse triangulation $\mathcal{T}_H$ and fine triangulation $\mathcal{T}_h$. (b) Nonoverlapping domain decomposition $\{ \Omega_k \}_{k=1}^N$. (c) Overlapping domain decomposition $\{ \Omega_k ' \}_{k=1}^N$.
• Figure 2: Reference solutions of the $p$-Laplacian problem \ref{['pLap_FEM']} ($p \in \{ 1.05, 1.1, 1.5, 5, 10, 20 \}$) computed by the adaptive Newton method Mishchenko:2023 ($h = 2^{-5}$).
• Figure 3: Decay of the relative energy error \ref{['rel_error']} in the two-level additive Schwarz method (\ref{['Alg:ASM']}) for the $p$-Laplacian problem \ref{['pLap_FEM']} ($p = 1.05$). Parameters $h$, $H$, and $\delta$ stand for the characteristic element size, subdomain size, and overlapping width among subdomains, respectively ($H/h = 2^3$).
• Figure 4: Decay of the relative energy error \ref{['rel_error']} in the two-level additive Schwarz method (\ref{['Alg:ASM']}) for the $p$-Laplacian problem \ref{['pLap_FEM']} ($p = 1.1$). Parameters $h$, $H$, and $\delta$ stand for the characteristic element size, subdomain size, and overlapping width among subdomains, respectively ($H/h = 2^3$).
• Figure 5: Decay of the relative energy error \ref{['rel_error']} in the two-level additive Schwarz method (\ref{['Alg:ASM']}) for the $p$-Laplacian problem \ref{['pLap_FEM']} ($p = 1.5$). Parameters $h$, $H$, and $\delta$ stand for the characteristic element size, subdomain size, and overlapping width among subdomains, respectively ($H/h = 2^3$).

Theorems & Definitions (49)

• Lemma 2.1
• Lemma 2.2
• Proof 1
• Proposition 2.3
• Proof 2
• Theorem 2.4
• Lemma 3.1
• Proof 3
• Lemma 3.2
• Proof 4