Divergence constraints are present in the governing equations of many physical phenomena, and they usually lead to a Poisson equation whose solution typically is the main bottleneck of many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, responsible for damping high-frequency error components, and the coarse-grid correction, which in turn reduces low-frequency modes. This work presents several strategies to make AMG more compute-intensive by leveraging reflection, translational and rotational symmetries, often present in academic and industrial configurations. The best-performing method, AMGR, is based on a multigrid reduction framework that introduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup and application costs of the top-level smoother. While preserving AMG's excellent convergence, AMGR allows replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.

翻译：散度约束存在于许多物理现象的控制方程中，通常会导致泊松方程，其求解往往是众多仿真代码的主要性能瓶颈。代数多重网格（AMG）无疑是求解泊松方程最强大的预条件子，其有效性源于平滑算子和粗网格校正的互补作用：平滑算子负责抑制高频误差分量，而粗网格校正则降低低频模态。本研究提出了几种策略，通过利用学术和工业配置中常见的反射、平移和旋转对称性，使AMG计算更加密集。性能最佳的方法AMGR基于多网格约简框架，该框架对多网格层次结构引入了激进的粗化策略，从而减少了顶层平滑算子的内存占用、设置和应用成本。在保持AMG优异收敛性的同时，AMGR允许用计算更密集的稀疏矩阵-矩阵乘积替代标准的稀疏矩阵-向量乘积，从而实现了显著的加速。在工业计算流体动力学应用中的数值实验表明，使用AMGR而非AMG求解泊松方程可获得高达70%的加速比。此外，强可扩展性和弱可扩展性分析均未显示出显著的性能退化。