Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems

TL;DR

This paper introduces a new constrained algebraic multigrid method, CℓAIR, that effectively solves both advection and diffusion dominated problems with scalable convergence and low complexity hierarchies.

Contribution

It generalizes the ℓAIR framework by integrating mode constraints, enhancing performance in diffusion regimes and enabling aggressive coarsening.

Findings

Achieves fast scalable convergence for advective and diffusive problems.

Maintains low complexity hierarchies in diffusion regimes.

Improves robustness of reduction-based AMG methods.

Abstract

This paper focuses on developing a reduction-based algebraic multigrid method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based algebraic multigrid (AMG) approach, $\ell$AIR (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion dominated problems in two or three dimensions. Motivated by the success of $\ell$AIR in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy minimization AMG methods with the local approximation of ideal operators used in $\ell$AIR. The resulting constrained $\ell$AIR (C$\ell$AIR) algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that has been previously difficult for reduction-based methods.