Comparative Convergence Analysis of Nonlinear AMLI-cycle Multigrid

TL;DR

This paper provides a comprehensive convergence analysis of the nonlinear AMLI-cycle multigrid method for symmetric positive definite problems, demonstrating uniform convergence and superior performance over V-cycle methods under certain conditions.

Contribution

It offers the first detailed convergence analysis of nonlinear AMLI-cycle multigrid methods, establishing uniform convergence and performance guarantees based on minimal assumptions.

Findings

The nonlinear AMLI-cycle MG method is uniformly convergent.

Under basic assumptions, it outperforms or matches the V-cycle MG method.

Numerical experiments confirm theoretical convergence results.

Abstract

The main purpose of this paper is to provide a comprehensive convergence analysis of nonlinear AMLI-cycle multigrid method for symmetric positive definite problems. Based on classical assumptions for approximation and smoothing properties, we show that the nonlinear AMLI-cycle MG method is uniformly convergent. Furthermore, under only the assumption that the smoother is convergent, we show that the nonlinear AMLI-cycle method is always better (or not worse) than the respective V-cycle MG method. Finally, numerical experiments are presented to illustrate the theoretical results.