Uniform preconditioners for the time dependent Stokes problem

Implicit time stepping procedures for the time dependent Stokes problem lead to stationary singular perturbation problems at each time step. These singular perturbation problems are systems of saddle point type, which formally approach a mixed formulation of the Poisson equation as the time step tends to zero. Preconditioners for discrete analogous of these systems are discussed. The preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization. The construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system.

[1]  Jinchao Xu,et al.  The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids , 1996, Computing.

[2]  H. Elman,et al.  Block preconditioners for the discrete incompressible Navier–Stokes equations , 2002 .

[3]  G. Wittum.  Multi-grid methods for stokes and navier-stokes equations , 1989 .

[4]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

[5]  Maxim A. Olshanskii,et al.  Navier-Stokes Equations in Rotation Form: A Robust Multigrid Solver for the Velocity Problem , 2002, SIAM J. Sci. Comput..

[6]  Rolf Stenberg,et al.  Error analysis of some nite element methods for the Stokes problem , 1990 .

[7]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[8]  Stefan Turek,et al.  Efficient Solvers for Incompressible Flow Problems - An Algorithmic and Computational Approach , 1999, Lecture Notes in Computational Science and Engineering.

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

[10]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[11]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[12]  R. Glowinski,et al.  Incompressible Computational Fluid Dynamics: On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow , 1993 .

[13]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[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]  Axel Klawonn,et al.  An Optimal Preconditioner for a Class of Saddle Point Problems with a Penalty Term , 1995, SIAM J. Sci. Comput..

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

[17]  L. Pavarino.  Indefinite overlapping Schwarz methods for time-dependent Stokes problems☆ , 2000 .

[18]  A. Wathen,et al.  Schur complement preconditioners for the Navier–Stokes equations , 2002 .

[19]  D. Arnold,et al.  Preconditioning discrete approximations of the Reissner-Mindlin plate model , 1997 .

[20]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[21]  O. Pironneau.  The Finite Element Method for Hyperbolic Systems , 1988 .

[22]  Anthony T. Patera,et al.  Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations , 1993, SIAM J. Sci. Comput..

[23]  Dietrich Braess,et al.  A multigrid method for a parameter dependent problem in solid mechanics , 1990 .

[24]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[25]  Xue-Cheng Tai,et al.  A Robust Finite Element Method for Darcy-Stokes Flow , 2002, SIAM J. Numer. Anal..

[26]  Ragnar Winther,et al.  A domain embedding preconditioner for the Lagrange multiplier system , 2000, Math. Comput..

[27]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

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

[29]  Panayot S. Vassilevski,et al.  Interior penalty preconditioners for mixed finite element approximations of elliptic problems , 1996, Math. Comput..

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

[31]  J. Cahouet,et al.  Some fast 3D finite element solvers for the generalized Stokes problem , 1988 .

[32]  Monique Dauge,et al.  Stationary Stokes and Navier-Stokes systems on two-or three-dimensional domains with corners , 1989 .

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

[34]  A. Wathen,et al.  Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners , 1994 .

[35]  H. Elman.  Preconditioners for saddle point problems arising in computational fluid dynamics , 2002 .