Convergence analysis of two-grid methods for symmetric positive semidefinite systems

TL;DR

This paper provides a new convergence analysis for two-grid methods applied to singular, symmetric positive semidefinite systems, introducing a concise identity and estimates that do not rely on additional matrix assumptions.

Contribution

It introduces a novel convergence identity and estimates for two-grid methods using Moore--Penrose inverse, applicable without extra assumptions on the coefficient matrix.

Findings

Derived a concise convergence identity for two-grid methods.

Established convergence estimates with approximate coarse solvers.

Applicable to singular, symmetric positive semidefinite systems without extra matrix assumptions.

Abstract

Two-grid theory plays a fundamental role in the design and analysis of multigrid methods. This paper is devoted to a new convergence analysis of two-grid methods for singular and symmetric positive semidefinite systems. Specifically, we derive a concise identity for characterizing the convergence factor of two-grid methods, with the Moore--Penrose inverse of coarse-grid matrix being used as a coarse solver. Furthermore, we present a convergence estimate for two-grid methods with approximate coarse solvers. Our new theory does not require any additional assumptions on the coefficient matrix, especially on its null space.