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

TL;DR

This paper develops a theoretical framework for analyzing the convergence of inexact two-grid methods, providing bounds and a unified theory that accommodates approximate solutions on the coarsest grid, enhancing multigrid analysis.

Contribution

It introduces a novel convergence analysis framework for inexact two-grid methods, extending existing theory to include approximate coarse-grid solutions.

Findings

Two-sided bounds for the error propagation matrix are established.

The framework recovers the exact two-grid convergence identity.

Unified convergence theory for multigrid methods with approximate coarse solutions is developed.

Abstract

Multigrid is one of the most efficient methods 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 motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately.