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.

翻译：本文聚焦于开发一种基于约化的代数多重网格方法，该方法适用于求解一般（非）对称线性系统，并且从纯对流到纯扩散问题均天然具有鲁棒性。初始动机源于一种新型基于约化的代数多重网格（AMG）方法——ℓAIR（局部近似理想限制），该方法专为解决对流主导问题而设计。尽管这一新求解器在对流主导区域非常有效，其性能在扩散占优情况下会下降。这与一般事实相符：基于约化的AMG方法通常随着问题规模增大，在复杂度增长和/或收敛速率方面表现欠佳，尤其对于二维或三维的扩散主导问题。受ℓAIR在对流区域成功应用的启发，本文旨在推广AIR框架，以改善求解器在扩散主导区域的性能。为此，我们提出了一种新颖方法，将能量最小化AMG方法中常用的模态约束与ℓAIR中使用的理想算子局部近似相结合。由此产生的约束ℓAIR（CℓAIR）算法能够在对流和扩散问题上实现快速可扩展的收敛。此外，通过激进粗化策略，该算法还能在扩散区域建立标准的低复杂度层次结构，而这在以往对基于约化的方法而言是难以实现的。