A sharp bound on the convergence rate of an aggregation-based algebraic multi-grid method applied to a 1D model problem

TL;DR

This paper establishes a precise lower bound on the convergence rate of an aggregation-based algebraic multigrid method applied to a 1D Poisson problem, demonstrating its effectiveness in error reduction.

Contribution

It provides a sharp theoretical bound on the convergence rate of a specific algebraic multigrid method for a 1D model problem, which was previously not well-understood.

Findings

The method reduces the A-norm of the error by at least 1/√2 per iteration.

The analysis applies to a standard 1D Poisson problem with Dirichlet boundary conditions.

The results offer insights into the efficiency of aggregation-based algebraic multigrid methods.

Abstract

We consider the linear system Ax=b arising from one-dimensional Poisson's equation with Dirichlet boundary conditions, where A is the square matrix with the stencil form [-1 2 -1]. Here we show that a pairwise aggregation-based algebraic 2-grid method reduces the A-norm of the error at each step by at least the factor $1/\sqrt{2}$.