Convergence analysis of two-level methods with general coarse solvers

TL;DR

This paper develops a general convergence framework for two-level multilevel methods that use inexact coarse solvers, including nonlinear and randomized approaches, enhancing practical applicability for large-scale PDE problems.

Contribution

It introduces a unified analysis framework for inexact two-level methods with diverse coarse solver types, extending convergence theory beyond exact Galerkin solutions.

Findings

Framework accommodates various solver types

Enables convergence analysis with approximate coarse solves

Supports nonlinear and randomized coarse solvers

Abstract

Multilevel methods are among the most efficient numerical methods for solving large-scale linear systems that arise from discretized partial differential equations. The fundamental module of such methods is a two-level procedure, which consists of compatible relaxation and coarse-level correction. Regarding two-level convergence theory, most previous works focus on the case of exact (Galerkin) coarse solver. In practice, however, it is often too costly to solve the Galerkin coarse-level system exactly when its size is relatively large. Compared with the exact case, the convergence theory of inexact two-level methods is of more practical significance, while it is still less developed in the literature, especially when nonlinear coarse solvers are used. In this paper, we establish a general framework for analyzing the convergence of inexact two-level methods, in which the coarse-level system is solved approximately by an inner iterative procedure. The framework allows us to use various (linear, nonlinear, deterministic, randomized, or hybrid) solvers in the inner iterations, as long as the corresponding accuracy estimates are available.