A unified approach to the design and analysis of AMG

TL;DR

This paper introduces a comprehensive framework for designing and analyzing two-level algebraic multigrid (AMG) methods, emphasizing the construction of optimal coarse spaces and providing convergence bounds.

Contribution

It presents a unified theoretical approach for AMG design, including the construction of coarse spaces and convergence analysis, with practical proof of uniform convergence for classical AMG.

Findings

Established a general framework for AMG analysis.

Proved uniform convergence of classical AMG for jump coefficient problems.

Provided bounds on convergence rates based on local Poincaré constants.

Abstract

In this work, we present a general framework for the design and analysis of two-level AMG methods. The approach is to find a basis for locally optimal or quasi-optimal coarse space, such as the space of constant vectors for standard discretizations of scalar elliptic partial differential equations. The locally defined basis elements are glued together using carefully designed linear extension maps to form a global coarse space. Such coarse spaces, constructed locally, satisfy global approximation property and by estimating the local Poincar{\' e} constants, we obtain sharp bounds on the convergence rate of the resulting two-level methods. To illustrate the use of the theoretical framework in practice, we prove the uniform convergence of the classical two level AMG method for finite element discretization of a jump coefficient problem on a shape regular mesh.