Navier-Stokes Equations in Rotation Form: A Robust Multigrid Solver for the Velocity Problem

The topic of this paper is motivated by the Navier--Stokes equations in rotation form. Linearization and application of an implicit time stepping scheme results in a linear stationary problem of Oseen type. In well-known solution techniques for this problem such as the Uzawa (or Schur complement) method, a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion-reaction type must be solved to update the velocity vector field. In this paper we analyze a standard finite element method for the discretization of this coupled system, and we introduce and analyze a multigrid solver for the discrete problem. Both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed. We prove stability results and discretization error bounds for the Galerkin finite element method. We present a convergence analysis of the multigrid method which shows the robustness of the solver. Results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver.

[1]  L. Wahlbin,et al.  On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions , 1983 .

[2]  M. Stynes,et al.  Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems , 1996 .

[3]  R. Codina,et al.  Finite element solution of the Stokes problem with dominating Coriolis force , 1997 .

[4]  G. Wittum,et al.  Downwind numbering: robust multigrid for convection—diffusion problems , 1997 .

[5]  H. Elman,et al.  Efficient preconditioning of the linearized Navier-Stokes , 1999 .

[6]  Thomas Probst,et al.  Downwind Gauß-Seidel Smoothing for Convection Dominated Problems , 1997, Numer. Linear Algebra Appl..

[7]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[8]  L. Wahlbin,et al.  Local behavior in finite element methods , 1991 .

[9]  Achi Brandt,et al.  Accelerated Multigrid Convergence and High-Reynolds Recirculating Flows , 1993, SIAM J. Sci. Comput..

[10]  Achi Brandt,et al.  Fast Multigrid Solution of the Advection Problem with Closed Characteristics , 1998, SIAM J. Sci. Comput..

[11]  A. Ramage.  A multigrid preconditioner for stabilised discretisations of advection-diffusion problems , 1999 .

[12]  Arnold Reusken,et al.  Fourier analysis of a robust multigrid method for convection-diffusion equations , 1995 .

[13]  M. Olshanskii,et al.  Stable finite‐element calculation of incompressible flows using the rotation form of convection , 2002 .

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

[15]  M. Olshanskii.  An iterative solver for the Oseen problem and numerical solution of incompressible Navier-Stokes equations , 1999 .

[16]  Joseph E. Pasciak,et al.  Uzawa type algorithms for nonsymmetric saddle point problems , 2000, Math. Comput..

[17]  S. Turek.  Efficient solvers for incompressible flow problems: An algorithmic approach . . , 1998 .

[18]  Wolfgang Hackbusch,et al.  Downwind Gauß‐Seidel Smoothing for Convection Dominated Problems , 1997 .

[19]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

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

[21]  C. Vreugdenhil,et al.  Numerical methods for advection-diffusion problems , 1993 .

[22]  M. Olshanskii.  A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods , 2002 .

[23]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .