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Multigrid analysis for the time dependent Stokes problem

Author: Maxim A. Olshanskii
Journal: Math. Comp. 81 (2012), 57-79
MSC (2010): Primary 65N55, 65N30, 65N15, 65F10
Published electronically: May 23, 2011
MathSciNet review: 2833487
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Abstract: Certain implicit time stepping procedures for the incompressible Stokes or Navier-Stokes equations lead to a singular-perturbed Stokes type problem at each time step. The paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method. Several smoothing iterative methods are considered: a smoother based on distributive iterations, the Braess-Sarazin and inexact Uzawa smoother. Convergence analysis is based on smoothing and approximation properties in special norms. A robust (independent of time step and mesh parameter) estimate is proved for the two-grid and multigrid W-cycle convergence factors.

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Additional Information

Maxim A. Olshanskii
Affiliation: Department of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Moscow 119899, Russia

Received by editor(s): April 29, 2009
Received by editor(s) in revised form: October 20, 2010
Published electronically: May 23, 2011
Additional Notes: The author was partially supported through the RFBR Grant 11-01-00767 and 09-01-00115
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.