We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations. However, in three-dimensional cases, the additive smoother outperforms its multiplicative counterpart due to the latter's lower potential for parallelism. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 percent.

翻译：本文讨论顶点块光滑子作为四阶椭圆型偏微分方程的重叠区域分解方法。我们证明该方法具有极高的数值效率并能产生高收敛率。此外，我们探讨了其高效实现的低秩张量近似技术。实验结果表明，非精确局部求解器产生的算法收敛速度快，且收敛性关于网格细化具有一致性。乘法光滑子在求解效率方面表现更优，所需迭代次数更少。然而在三维情形中，由于乘法光滑子的并行潜力较低，加法光滑子反而展现出更佳性能。此外，求解器架构支持混合精度计算方案：在单精度下执行多重网格预处理器，同时在双精度下执行外层迭代，从而使计算吞吐量提升最高达70%。