Comparative Convergence Analysis of Nonlinear AMLI-Cycle Multigrid

Abstract

The main purpose of this paper is to provide a comprehensive convergence analysis of the nonlinear algebraic multilevel iteration (AMLI)-cycle multigrid (MG) 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.