Preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and Stokes problems

Abstract Mixed spectral methods applied to elasticity and Stokes problems in three dimensions produce large symmetric indefinite systems of equations. These systems are sparse and more ill-conditioned than those produced by h -version finite element methods. These stiffness matrices have the structure of saddle point problems with a penalty term. In this paper, preconditioned iterative methods of conjugate residual type are introduced for these systems. The convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n . Several numerical experiments on three dimensional model problems confirm the results obtained.

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

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

[3]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

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

[5]  Michel O. Deville,et al.  Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems , 1990, SIAM J. Sci. Comput..

[6]  K. Bathe,et al.  The inf-sup test , 1993 .

[7]  H. Elman.  Multigrid and Krylov subspace methods for the discrete Stokes equations , 1994 .

[8]  R. Stenberg,et al.  Mixed $hp$ finite element methods for problems in elasticity and Stokes flow , 1996 .

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

[10]  Ivo Babuska,et al.  The p and h-p Versions of the Finite Element Method, Basic Principles and Properties , 1994, SIAM Rev..

[11]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[12]  Jan Mandel,et al.  Iterative methods for p-version finite elements: preconditioning thin solids , 1996 .

[13]  Wagdi G. Habashi,et al.  Solution techniques for large-scale CFD problems , 1995 .

[14]  Paul Fischer,et al.  Spectral element methods for large scale parallel Navier—Stokes calculations , 1994 .

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

[16]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

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

[18]  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..

[19]  Jan Mandel,et al.  An iterative solver for p-version finite elements in three dimensions , 1994 .

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

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

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

[23]  Olof B. Widlund,et al.  Preconditioned conjugate gradient solvers for spectral elements in 3D , 1995 .

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

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

[26]  I. Babuska,et al.  Locking effects in the finite element approximation of elasticity problems , 1992 .

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