Refined saddle-point preconditioners for discretized Stokes problems

This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online.

[1]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[2]  Z. Strakos,et al.  Krylov Subspace Methods: Principles and Analysis , 2012 .

[3]  Howard C. Elman,et al.  IFISS: A Computational Laboratory for Investigating Incompressible Flow Problems , 2014, SIAM Rev..

[4]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[5]  Sander Rhebergen,et al.  Analysis of Block Preconditioners for Models of Coupled Magma/Mantle Dynamics , 2013, SIAM J. Sci. Comput..

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

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

[8]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[9]  R. Varga,et al.  Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods , 1961 .

[10]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

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

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

[13]  Robert L. Lee,et al.  The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2 , 1981 .

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

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

[16]  D. May,et al.  Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics , 2008 .

[17]  C. Dohrmann,et al.  A stabilized finite element method for the Stokes problem based on polynomial pressure projections , 2004 .

[18]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[19]  Michele Benzi,et al.  On the eigenvalues of a class of saddle point matrices , 2006, Numerische Mathematik.

[20]  Owe Axelsson Eigenvalue Estimates for Preconditioned Saddle Point Matrices , 2003, LSSC.

[21]  A. Wathen,et al.  The convergence rate of the minimal residual method for the Stokes problem , 1995 .

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

[23]  YU. A. KUZNETSOV,et al.  Efficient iterative solvers for elliptic finite element problems on nonmatching grids , 1995 .

[24]  A. Wathen,et al.  Chebyshev semi-iteration in preconditioning for problems including the mass matrix. , 2008 .

[25]  Howard C. Elman,et al.  Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow , 2007, TOMS.

[26]  Markus Neher,et al.  Complex standard functions and their implementation in the CoStLy library , 2007, TOMS.

[27]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.