On the eigenvalues of a class of saddle point matrices

We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners.

[1]  G. Golub,et al.  Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems , 2003 .

[2]  Kent-André Mardal,et al.  Uniform preconditioners for the time dependent Stokes problem , 2006, Numerische Mathematik.

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

[4]  R. Temam.  Navier-Stokes Equations , 1977 .

[5]  Avram Sidi.  A Zero-Cost Preconditioning for a Class of Indefinite Linear Systems , 2003 .

[6]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

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

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

[9]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[10]  K. Hannabuss,et al.  MATRICES AND INDEFINITE SCALAR PRODUCTS (Operator Theory: Advances and Applications, 8) , 1984 .

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

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[14]  Paul E. Saylor,et al.  The Role of the Inner Product in Stopping Criteria for Conjugate Gradient Iterations , 2001 .

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

[16]  R. Freund,et al.  Software for simplified Lanczos and QMR algorithms , 1995 .

[17]  R. Glowinski.  Finite element methods for incompressible viscous flow , 2003 .

[18]  Gene H. Golub,et al.  A Preconditioner for Generalized Saddle Point Problems , 2004, SIAM J. Matrix Anal. Appl..

[19]  V. Simoncini,et al.  Block--diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 1999 .

[20]  Leiba Rodman,et al.  Matrices and indefinite scalar products , 1983 .

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

[22]  Franz Peherstorfer,et al.  CHEBYSHEV APPROXIMATION VIA POLYNOMIAL MAPPINGS AND THE CONVERGENCE BEHAVIOUR OF KRYLOV SUBSPACE METHODS , 2001 .

[23]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[24]  Gene H. Golub,et al.  On Solving Block-Structured Indefinite Linear Systems , 2003, SIAM J. Sci. Comput..

[25]  G. W. Stewart,et al.  Matrix algorithms , 1998 .

[26]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[27]  G. Stewart.  Matrix Algorithms, Volume II: Eigensystems , 2001 .

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

[29]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[30]  O. Axelsson.  Iterative solution methods , 1995 .

[31]  P. G. Ciarlet,et al.  Numerical Methods for Fluids, Part 3 , 2003 .

[32]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

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

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

[35]  Michele Benzi,et al.  Spectral Properties of the Hermitian and Skew-Hermitian Splitting Preconditioner for Saddle Point Problems , 2005, SIAM J. Matrix Anal. Appl..