Abstract Algebraic multigrid (AMG) is an efficient iterative method for solving linear equation systems arising from the elliptic partial differential equations. The coarsening algorithm, which determines the coarse-variable set in the classical AMG, is a critical component. This paper targets at reducing the overall solution time of the classical AMG by improving the quality of the coarse-variable set obtained by the coarsening algorithm. We combine the classical coarsening algorithm with the compatible relaxation (CR)-based coarsening algorithm to construct the coarse-variable set. The combined coarsening algorithm constructs the coarse-variable set within two stages. In the first stage, a basic coarse-variable set is built by the classical coarsening algorithm, e.g., PMIS. In the second stage, the quality of the set is measured based on compatible relaxation, and the variables that converge slowly in the CR relaxation are added into the previous set. We test various model problems, as well as some linear equation systems arising from real applications, to verify the effectiveness of our method.

中文翻译：

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代数多重网格粗化的一种补充策略

摘要 代数多重网格(AMG)是求解由椭圆偏微分方程产生的线性方程组的一种有效迭代方法。确定经典 AMG 中的粗变量集的粗化算法是一个关键组件。本文旨在通过提高粗化算法获得的粗变量集的质量来减少经典 AMG 的整体求解时间。我们将经典的粗化算法与基于兼容松弛（CR）的粗化算法相结合来构造粗变量集。组合粗化算法在两个阶段内构造粗变量集。在第一阶段，通过经典的粗化算法，例如PMIS，建立一个基本的粗变量集。在第二阶段，集合的质量是基于相容松弛来衡量的，在 CR 松弛中收敛缓慢的变量被添加到之前的集合中。我们测试了各种模型问题，以及一些实际应用中产生的线性方程组，以验证我们方法的有效性。