The solution of saddle-point problems, such as the Stokes equations, is a challenging task, especially in large-scale problems. Multigrid methods are one of the most efficient solvers for such systems of equations and can achieve convergence rates independent of the problem size. The smoother is a crucial component of multigrid methods and significantly affects its overall efficiency. We propose a Vanka-type smoother that we refer to as Restricted Additive Vanka and investigate its convergence in the context of adaptive geometric multigrid methods for the Stokes equations. The proposed smoother has the advantage of being an additive method and provides favorable properties in terms of algorithmic complexity, scalability and applicability to high-performance computing. We compare the performance of the smoother with two variants of the classical Vanka smoother using numerical benchmarks for the Stokes problem. We find that the restricted additive smoother achieves comparable convergence rates to the classical multiplicative Vanka smoother while being computationally less expensive per iteration, which results in faster solution runtimes.
翻译:诸如斯托克斯方程式等马鞍点问题的解决办法是一项艰巨的任务,特别是在大规模问题中。多格格丽德方法是这类方程式系统最有效的解决办法之一,能够实现与问题大小无关的趋同率。光滑是多格格方法的一个关键组成部分,并极大地影响其总体效率。我们建议使用一个Vanka型平滑器,我们称之为Restricted Additive Vanka, 并在斯托克斯方程式的适应性几何多格格方法中调查其趋同性。提议的平滑器的优点是是一种添加法,在算法的复杂性、可缩放性和对高性计算的适用性方面提供了有利的特性。我们用用于斯托克斯问题的数字基准的古典Vanka光滑的两种变式比较了光滑的性能。我们发现,有限的添加法滑动剂在计算每升速成本较低的情况下达到与古典多法式Vanka平滑的相相近的趋同率,而计算成本较低,从而导致更快的溶性运行时间。