Multigrid and Krylov subspace methods for the discrete Stokes equations

SUMMARY Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizatiom a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the dkretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.

[1]  Seymour V. Parter,et al.  On “two-line” iterative methods for the Laplace and biharmonic difference equations , 1959, Numerische Mathematik.

[2]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[3]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[4]  S. Eisenstat,et al.  The Modified Conjugate Residual Method for Partial Differential Equations. , 1977 .

[5]  A. Brandt,et al.  Multigrid Solutions to Elliptic Flow Problems , 1979 .

[6]  R. Glowinski Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[7]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

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

[9]  F. Brezzi,et al.  On the Stabilization of Finite Element Approximations of the Stokes Equations , 1984 .

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

[11]  R. Verfürth A combined conjugate gradient - multi-grid algorithm for the numerical solution of the Stokes problem , 1984 .

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

[13]  Juhani Pitkäranta,et al.  A multigrid version of a simple finite element method for the Stokes problem , 1985 .

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

[15]  A. Wathen Realistic Eigenvalue Bounds for the Galerkin Mass Matrix , 1987 .

[16]  R. Sani,et al.  On pressure boundary conditions for the incompressible Navier‐Stokes equations , 1987 .

[17]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[18]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[19]  R. Bank,et al.  A class of iterative methods for solving saddle point problems , 1989 .

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

[21]  Gabriel Wittum,et al.  On the convergence of multi-grid methods with transforming smoothers , 1990 .

[22]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .

[23]  A. Niestegge,et al.  Analysis of a multigrid strokes solver , 1990 .

[24]  Oliver A. McBryan,et al.  Normalized Convergence Rates for the PSMG Method , 1991, SIAM J. Sci. Comput..

[25]  Naomi H. Decker,et al.  Note on the Parallel Efficiency of the Frederickson-McBryan Multigrid Algorithm , 1990, SIAM J. Sci. Comput..

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

[27]  P. Wesseling An Introduction to Multigrid Methods , 1992 .

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

[29]  Andrew J. Wathen,et al.  Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners , 1993 .

[30]  A. Wathen,et al.  FAST ITERATIVE SOLUTION OF STABILIZED STOKES SYSTEMS .1. USING SIMPLE DIAGONAL PRECONDITIONERS , 1993 .

[31]  A. Wathen,et al.  Iterative solution techniques for the stokes and Navier‐Stokes equations , 1994 .

[32]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[33]  A. Brandt Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L 2 -norm , 1994 .

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

[35]  David J. Silvester Optimal low order finite element methods for incompressible flow , 1994 .

[36]  X. Wu,et al.  Analysis and convergence of the MAC scheme. II. Navier-Stokes equations , 1996, Math. Comput..