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A local Fourier analysis of additive Vanka relaxation for the Stokes equations
Author(s) -
Farrell Patrick E.,
He Yunhui,
MacLachlan Scott P.
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2306
Subject(s) - multigrid method , discretization , relaxation (psychology) , convergence (economics) , grid , finite element method , mathematics , mathematical optimization , fourier transform , fourier analysis , fast fourier transform , partial differential equation , algorithm , computer science , mathematical analysis , geometry , psychology , social psychology , physics , economics , thermodynamics , economic growth
Summary 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.