Multigrid analysis for the time dependent Stokes problem

Olshanskii, Maxim

doi:10.1090/S0025-5718-2011-02494-4

Multigrid analysis for the time dependent Stokes problem
HTML articles powered by AMS MathViewer

by Maxim A. Olshanskii;

Math. Comp. 81 (2012), 57-79

DOI: https://doi.org/10.1090/S0025-5718-2011-02494-4

Published electronically: May 23, 2011

PDF | Request permission

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.

References

D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337–344 (1985). MR 799997, DOI 10.1007/BF02576171
Randolph E. Bank, Bruno D. Welfert, and Harry Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990), no. 7, 645–666. MR 1031439, DOI 10.1007/BF01405194
Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005), 1–137. MR 2168342, DOI 10.1017/S0962492904000212
M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33 (1979), no. 2, 211–224. MR 549450, DOI 10.1007/BF01399555
Dietrich Braess and Wolfgang Dahmen, A cascadic multigrid algorithm for the Stokes equations, Numer. Math. 82 (1999), no. 2, 179–191. MR 1685458, DOI 10.1007/s002110050416
D. Braess and R. Sarazin, An efficient smoother for the Stokes problem, Appl. Numer. Math. 23 (1997), no. 1, 3–19. Multilevel methods (Oberwolfach, 1995). MR 1438078, DOI 10.1016/S0168-9274(96)00059-1
J. H. Bramble and J. E. Pasciak, Iterative techniques for time dependent Stokes problems, Comput. Math. Appl. 33 (1997), no. 1-2, 13–30. Approximation theory and applications. MR 1442058, DOI 10.1016/S0898-1221(96)00216-7
Susanne C. Brenner, A nonconforming multigrid method for the stationary Stokes equations, Math. Comp. 55 (1990), no. 192, 411–437. MR 1035927, DOI 10.1090/S0025-5718-1990-1035927-5
James H. Bramble, Joseph E. Pasciak, and Olaf Steinbach, On the stability of the $L^2$ projection in $H^1(\Omega )$, Math. Comp. 71 (2002), no. 237, 147–156. MR 1862992, DOI 10.1090/S0025-5718-01-01314-X
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237
Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489, DOI 10.1137/0520006
L. Durlofsky and J. F. Brady, Analysis of the Brinkman equation as a model for flow in porous media, Phys. Fluids, 30 (1987), pp. 3329–3342.
H.C. Elman, Multigrid and Krylov subspace methods for the discrete Stokes equations, Int. J. Numer. Methods Fluids, 22 (1998), pp. 755 – 770.
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
W. Hackbusch, Multi-grid Methods and Applications, Berlin, Heidelberg: Springer, 1985.
Yu. Iliash, R. Tuomo, and J. Toivanen, Two iterative methods for solving the Stokes problem, Laboratory of Scientific Computing, University of Jyvaskyla, Report 2, 1993.
Georgij M. Kobelkov and Maxim A. Olshanskii, Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math. 86 (2000), no. 3, 443–470. MR 1785417, DOI 10.1007/s002110000160
Maxim Larin and Arnold Reusken, A comparative study of efficient iterative solvers for generalized Stokes equations, Numer. Linear Algebra Appl. 15 (2008), no. 1, 13–34. MR 2388412, DOI 10.1002/nla.561
Kent Andre Mardal, Xue-Cheng Tai, and Ragnar Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal. 40 (2002), no. 5, 1605–1631. MR 1950614, DOI 10.1137/S0036142901383910
S. Manservisi, Numerical analysis of Vanka-type solvers for steady Stokes and Navier-Stokes flows, SIAM J. Numer. Anal. 44 (2006), no. 5, 2025–2056. MR 2263039, DOI 10.1137/060655407
Kent-Andre Mardal and Ragnar Winther, Uniform preconditioners for the time dependent Stokes problem, Numer. Math. 98 (2004), no. 2, 305–327. MR 2092744, DOI 10.1007/s00211-004-0529-6
M. A. Olshanskii and A. Reusken, On the convergence of a multigrid method for linear reaction-diffusion problems, Computing 65 (2000), no. 3, 193–202. MR 1810324, DOI 10.1007/s006070070006
Maxim A. Olshanskii, Jörg Peters, and Arnold Reusken, Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations, Numer. Math. 105 (2006), no. 1, 159–191. MR 2257388, DOI 10.1007/s00211-006-0031-4
Torgeir Rusten, Panayot S. Vassilevski, and Ragnar Winther, Interior penalty preconditioners for mixed finite element approximations of elliptic problems, Math. Comp. 65 (1996), no. 214, 447–466. MR 1333325, DOI 10.1090/S0025-5718-96-00720-X
V. V. Shaĭdurov, Mnogosetochnye metody konechnykh èlementov, “Nauka”, Moscow, 1989 (Russian). MR 1007306
Joachim Schöberl and Walter Zulehner, On Schwarz-type smoothers for saddle point problems, Numer. Math. 95 (2003), no. 2, 377–399. MR 2001083, DOI 10.1007/s00211-002-0448-3
David Silvester and Andrew Wathen, Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), no. 5, 1352–1367. MR 1293519, DOI 10.1137/0731070
Rob Stevenson, Nonconforming finite elements and the cascadic multi-grid method, Numer. Math. 91 (2002), no. 2, 351–387. MR 1900924, DOI 10.1007/s002110100344
Zhanye Tong and Ahmed Sameh, On an iterative method for saddle point problems, Numer. Math. 79 (1998), no. 4, 643–646. MR 1635705, DOI 10.1007/s002110050356
Stefan Turek, Efficient solvers for incompressible flow problems, Lecture Notes in Computational Science and Engineering, vol. 6, Springer-Verlag, Berlin, 1999. An algorithmic and computational approach; With 1 CD-ROM (“Virtual Album”: UNIX/LINUX, Windows, POWERMAC; “FEATFLOW 1.1”: UNIX/LINUX). MR 1691839, DOI 10.1007/978-3-642-58393-3
S. P. Vanka, Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comput. Phys. 65 (1986), no. 1, 138–158. MR 848451, DOI 10.1016/0021-9991(86)90008-2
R. Verfürth, A multilevel algorithm for mixed problems, SIAM J. Numer. Anal. 21 (1984), no. 2, 264–271. MR 736330, DOI 10.1137/0721019
P. Wesseling and C. W. Oosterlee, Geometric multigrid with applications to computational fluid dynamics, J. Comput. Appl. Math. 128 (2001), no. 1-2, 311–334. Numerical analysis 2000, Vol. VII, Partial differential equations. MR 1820879, DOI 10.1016/S0377-0427(00)00517-3
Xiaoping Xie, Jinchao Xu, and Guangri Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math. 26 (2008), no. 3, 437–455. MR 2421892
J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing 56 (1996), no. 3, 215–235 (English, with English and German summaries). International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994). MR 1393008, DOI 10.1007/BF02238513
W. Zulehner, A class of smoothers for saddle point problems, Computing 65 (2000), no. 3, 227–246. MR 1810326, DOI 10.1007/s006070070008