This paper provides a rounding-error analysis for two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the coarse grid to allow for an approximate coarse-grid solve. Leveraging previously published results, this two-grid theory can then be extended to general $V(\mu,\nu)$-cycles, as well as full multigrid (FMG). It can also be extended to mixed-precision iterative refinement (IR) based on these cycles. An added benefit of the theory here over previous work is that it is obtained in a more organized, transparent, and simpler way.

翻译：本文对在粗化前后各使用一次松弛步骤的两网格方法进行了舍入误差分析。该分析基于浮点运算，并聚焦于一种在粗网格上引入扰动以允许近似粗网格求解的两网格策略。借助先前发表的研究成果，这一两网格理论可进一步推广至一般的$V(\mu,\nu)$循环以及完全多重网格（FMG），亦可拓展至基于此类循环的混合精度迭代精化（IR）方法。相较于已有工作，本文理论的一个额外优势在于其推导过程更加系统、透明且简洁。