A local Fourier analysis of additive Vanka relaxation for the Stokes equations
A local Fourier analysis of additive Vanka relaxation for the Stokes equations
复制标题
Stokes 方程的加性 Vanka 松弛的局部傅立叶分析
DOI：
--
复制
发表时间：
2019
影响因子：
4.3
通讯作者：
S. MacLachlan
中科院分区：
文献类型：
--
作者：
P. Farrell;Yunhui He;S. MacLachlan
多重网格法是许多离散偏微分方程组的常用求解算法，无论是作为独立的迭代求解器还是作为预条件算子，都具有很高的效率。然而，多重网格组件(如松弛方案和网格转移算子)的选择和优化是设计最优高效算法的关键。众所周知，局部傅里叶分析(LFA)是预测和分析这些部件性能的有用工具。本文给出了Stokes方程混合有限元离散的基于加性Vanka松弛格式的整体多重网格方法的局部傅立叶分析。该分析提供了对Vanka松弛的“补丁”选择的洞察，揭示了较小的补丁为每个浮点运算提供了更有效的收敛。提出了最小化两个网格收敛因子的参数，并通过数值实验验证了LFA预测的有效性。
Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid‐transfer operators is crucial to the design of optimally efficient algorithms. It is well known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this article, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite‐element discretizations of the Stokes equations. The analysis offers insight into the choice of “patches” for the Vanka relaxation, revealing that smaller patches offer more effective convergence per floating point operation. Parameters that minimize the two‐grid convergence factor are proposed and numerical experiments are presented to validate the LFA predictions.