A new characterization of the convergence factor of two-level methods

TL;DR

This paper introduces a simple identity to characterize the convergence factor of two-level multilevel methods, applicable to overlapping or non-overlapping hierarchical spaces, aiding in their analysis and design.

Contribution

The paper presents a novel, concise identity for analyzing two-level method convergence, enhancing understanding and facilitating the development of multilevel algorithms.

Findings

Provides a new convergence characterization identity

Applies to overlapping and non-overlapping hierarchical spaces

Offers insights for designing multilevel methods

Abstract

Multilevel methods are among the most efficient numerical methods for solving large-scale systems of equations that arise from discretized partial differential equations. Two-level convergence theory plays a fundamental role in the analysis and design of multilevel methods. In this paper, we present a concise and easy-to-use identity for characterizing the convergence factor of two-level methods, whose hierarchical spaces can be either overlapping or non-overlapping. In order to illustrate its usability and convenience, we give several applications, which offer new insights into the design of multilevel methods.