Norm-based convergence bounds for nonsymmetric algebraic V-cycle multigrid methods

TL;DR

This paper extends a theoretical framework for analyzing nonsymmetric algebraic multigrid methods using HPD-induced norms, providing new convergence bounds and insights into multigrid operators and coarse-grid corrections.

Contribution

It generalizes existing analysis to any HPD matrix B, introduces new error operators, and extends convergence bounds to nonsymmetric V-cycle multigrid methods.

Findings

Provided sharp estimates for error propagation matrices.

Showed norms decrease with larger coarse spaces.

Extended the V-cycle convergence bound to nonsymmetric cases.

Abstract

Recently a new approach to analyze and create algebraic multigrid methods (AMG) for nonsymmetric and indefinite matrices was established. Convergence is measured in general norms induced by a certain HPD matrix $B$ and $B$-orthogonal projections built by compatible transfer operators are used. Here we continue our theoretical framework, started in Nabben and Rooch (2026), for nonsymmetric algebraic multigrid methods using any HPD matrix $B$ to induce a norm. Our framework not only includes all recent results but also provides many new results. We consider two, slightly different, multigrid operators. The first one is the natural generalization of the error operator in the HPD case. The second operator is simpler to apply and has been studied before. However, an additional condition for the smoother $M^{-1}A$ is needed, which is in our terminology the $B$-normality. We explain the differences and similarities of both operators in detail and show, why the extra condition is needed. We consider arbitrary interpolation and restriction operators that result in $B$-orthogonal coarse-grid corrections and give sharp estimates for the norm of the error propagation matrices for the two-grid methods. We also show, that the norms are decreasing if we increase the size of the coarse space. Moreover, we are able to extend the landmark $V$-cycle bound by McCormick to the nonsymmetric case.