Convergence analysis of a two-grid method for nonsymmetric positive definite problems

TL;DR

This paper provides a new convergence analysis for a two-grid method applied to nonsymmetric positive definite problems, introducing an identity for the convergence factor and exploring optimal restriction operators and inexact solvers.

Contribution

It introduces an elegant identity for the two-grid convergence factor in nonsymmetric problems and analyzes both exact and inexact coarse solvers, advancing multigrid theory.

Findings

Established an identity characterizing the convergence factor

Derived optimal restriction operators for improved convergence

Provided estimates for inexact coarse solvers

Abstract

Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.