This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two-level convergence even for systems of equations. This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.

翻译：本文介绍了一种新颖的方法,用于对来自某些椭圆二等分级部分差异方程的有限元素分解的大型线性方程系统的代数镜数多格法方法。根据由边缘和顶端贡献组成的离散能量,我们能够制定粗化标准,保证即使在等式系统中也达到两级趋同。这种能量还使我们能够用规定仍能准确保存内核矢量的聚变模式建造延长期。这使我们能够实现简单优化,简化平行状态,减少粗皮水平上的通信。数字实验显示了执行方法的效率和稳健性以及可伸缩性。