A Robust Multigrid Method for the Time-Dependent Stokes Problem

In the present paper we propose a coupled multigrid method for generalized Stokes flow problems. Such problems occur as subproblems in implicit time-stepping approaches for time-dependent Stokes problems. The discretized Stokes system is a large-scale linear system whose condition number depends on the grid size of the spatial discretization and of the length of the time step. Recently, for this problem a coupled multigrid method has been proposed, where in each smoothing step a Poisson problem has to be solved (approximately) for the pressure field. In the present paper, we propose a coupled multigrid method where the solution of such sub-problems is not needed. We prove that the proposed method shows robust convergence behavior in the grid size of the spatial discretization and of the length of the time-step.

[1]  Joachim Schöberl,et al.  On Schwarz-type Smoothers for Saddle Point Problems , 2003, Numerische Mathematik.

[2]  P. Wesseling,et al.  Geometric multigrid with applications to computational fluid dynamics , 2001 .

[3]  D. Braess,et al.  An efficient smoother for the Stokes problem , 1997 .

[4]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[5]  Stefan Takacs,et al.  Convergence Analysis of All-at-Once Multigrid Methods for Elliptic Control Problems under Partial Elliptic Regularity , 2013, SIAM J. Numer. Anal..

[6]  Susanne C. Brenner,et al.  Multigrid methods for parameter dependent problems , 1996 .

[7]  W. Marsden.  I and J , 2012 .

[8]  Maxim A. Olshanskii,et al.  Effective preconditioning of Uzawa type schemes for a generalized Stokes problem , 2000, Numerische Mathematik.

[9]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[10]  R. Kellogg,et al.  A regularity result for the Stokes problem in a convex polygon , 1976 .

[11]  R. Verfürth.  A Multilevel Algorithm for Mixed Problems , 1984 .

[12]  Stefan Takacs,et al.  A robust all-at-once multigrid method for the Stokes control problem , 2015, Numerische Mathematik.

[13]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[14]  Maxim A. Olshanskii,et al.  Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations , 2006, Numerische Mathematik.

[15]  S. Vanka.  Block-implicit multigrid solution of Navier-Stokes equations in primitive variables , 1986 .

[16]  Maxim A. Olshanskii,et al.  Multigrid analysis for the time dependent Stokes problem , 2012, Math. Comput..

[17]  O. Pironneau,et al.  Error estimates for finite element method solution of the Stokes problem in the primitive variables , 1979 .

[18]  Walter Zulehner,et al.  A Class of Smoothers for Saddle Point Problems , 2000, Computing.

[19]  J. Pasciak,et al.  Iterative techniques for time dependent Stokes problems , 1997 .

[20]  Rüdiger Verfürth,et al.  Error estimates for a mixed finite element approximation of the Stokes equations , 1984 .

[21]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[22]  Kent-André Mardal,et al.  Uniform preconditioners for the time dependent Stokes problem , 2004, Numerische Mathematik.

[23]  W. Zulehner.  Non-standard Norms and Robust Estimates for Saddle Point Problems , 2011 .