This article 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 such as linear elasticity . 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.

中文翻译：

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基于边矩阵辅助拓扑的弹性代数多重网格方法

本文介绍了一种用于大型线性方程组代数多重网格方法的新方法，这些线性方程组来自某些椭圆二阶偏微分方程的有限元离散化。基于由边缘和顶点贡献组成的离散能量，我们能够开发粗化标准，即使对于线性弹性等方程组也能保证两级收敛。这种能量还允许我们构建具有指定稀疏模式的扩展，同时仍然准确地保留内核向量。这些允许直接优化，简化并行化并减少粗略级别的通信。数值实验证明了该方法的效率和鲁棒性以及实现的可扩展性。