Core Concepts
The additive Schwarz methods for the p-Laplacian problem exhibit linear convergence, which is faster than the previously known sublinear convergence rates.
Abstract
The paper presents a novel convergence analysis for additive Schwarz methods applied to the p-Laplacian problem. While existing theoretical results have shown sublinear convergence rates for these methods, numerical experiments have demonstrated a linear convergence behavior.
The key insights are:
- The authors introduce the use of a quasi-norm that approximates the Bregman distance of the convex energy functional associated with the p-Laplacian problem. This allows for a tighter two-sided bound compared to using the standard Sobolev norm.
- A quasi-norm version of the Poincaré-Friedrichs inequality is derived, which plays a crucial role in establishing a quasi-norm stable decomposition for the two-level domain decomposition setting.
- By leveraging these key elements, the authors prove the linear convergence of the two-level additive Schwarz method for the p-Laplacian problem, bridging the gap between the theoretical and empirical results.
The linear convergence result is the first of its kind for additive Schwarz methods applied to the p-Laplacian problem.
Stats
The paper does not provide any explicit numerical data or statistics to support the claims. However, it references several previous works that have presented numerical experiments demonstrating the linear convergence behavior of additive Schwarz methods for the p-Laplacian problem.
Quotes
"While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate."
"We claim that the sublinear convergence rates given in the existing works [29, 38] are caused by this looseness."
"A novelty in this paper is that, by extending the idea of [29], a new convergence theory for additive Schwarz methods is obtained in terms of the quasi-norm, which utilizes (1.6) to obtain the linear convergence rate of additive Schwarz methods for the p-Laplacian."