Convergence analysis of inexact two-grid methods: A theoretical framework

Abstract

Multigrid methods are among the most efficient iterative techniques for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in understanding and designing multigrid methods. Convergence analysis of exact two-grid methods (i.e., the Galerkin coarse-grid system is solved exactly) has been well developed: the convergence factor of exact two-grid methods can be characterized by an identity. However, convergence theory of inexact ones (i.e., the coarse-grid problem is solved approximately) is still less mature. In this paper, a theoretical framework for the convergence analysis of inexact two-grid methods is developed. More specifically, two-sided bounds for the energy norm of the error propagation matrix are established under different approximation conditions, from which one can readily get the identity for the convergence factor of exact two-grid methods.