Spectral analysis of the preconditioned system for the 3 × 3 block saddle point problem

In this work, we consider some preconditioning techniques for a class of 3 × 3 block saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and some other applications. We propose an exact block diagonal preconditioner for solving the symmetric saddle point problem and its nonsymmetric form. We show that the corresponding preconditioned systems have six different eigenvalues. For the needs of practical application, we also present a class of inexact block diagonal preconditioners for solving the saddle point problems. For the symmetric system, we estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrix, respectively. For the nonsymmetric system, we derive some explicit and sharp bounds on the real and complex eigenvalues. Numerical experiments are presented to demonstrate the effectiveness and robustness of all these new preconditioners.

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

[2]  Jinchao Xu,et al.  Structure-preserving finite element methods for stationary MHD models , 2015, Math. Comput..

[3]  Michael K. Ng,et al.  New preconditioners for saddle point problems , 2006, Appl. Math. Comput..

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

[5]  Luca Bergamaschi,et al.  On eigenvalue distribution of constraint‐preconditioned symmetric saddle point matrices , 2012, Numer. Linear Algebra Appl..

[6]  Michael K. Ng,et al.  Constraint Preconditioners for Symmetric Indefinite Matrices , 2009, SIAM J. Matrix Anal. Appl..

[7]  Stefan A. Funken,et al.  Fast solvers with block‐diagonal preconditioners for linear FEM–BEM coupling , 2009, Numer. Linear Algebra Appl..

[8]  Z. Cao.  Positive stable block triangular preconditioners for symmetric saddle point problems , 2007 .

[9]  Pierre Degond,et al.  An analysis of the Darwin model of approximation to Maxwell’s equations , 1992 .

[10]  Chuanqing Gu,et al.  A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems , 2014, Appl. Math. Comput..

[11]  B. Fischer.  Polynomial Based Iteration Methods for Symmetric Linear Systems , 1996 .

[12]  Luca Bergamaschi,et al.  RMCP: Relaxed Mixed Constraint Preconditioners for Saddle Point Linear Systems arising in Geomechanics , 2012 .

[13]  Yang Cao,et al.  Shift-splitting preconditioners for saddle point problems , 2014, J. Comput. Appl. Math..

[14]  Changfeng Ma,et al.  A generalized shift-splitting preconditioner for saddle point problems , 2015, Appl. Math. Lett..

[15]  J. Zou,et al.  An effective preconditioner for a PML system for electromagnetic scattering problem , 2015 .

[16]  Yang Cao,et al.  On parameterized block triangular preconditioners for generalized saddle point problems , 2010, Appl. Math. Comput..

[17]  Peter Monk,et al.  Analysis of a finite element method for Maxwell's equations , 1992 .

[18]  Luca Bergamaschi,et al.  A note on eigenvalue distribution of constraint‐preconditioned symmetric saddle point matrices , 2014, Numer. Linear Algebra Appl..

[19]  Na Huang,et al.  An inexact relaxed DPSS preconditioner for saddle point problem , 2015, Appl. Math. Comput..

[20]  Eric de Sturler,et al.  Block-Diagonal and Constraint Preconditioners for Nonsymmetric Indefinite Linear Systems. Part I: Theory , 2005, SIAM J. Sci. Comput..

[21]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[22]  Piotr Krzyzanowski,et al.  On block preconditioners for saddle point problems with singular or indefinite (1, 1) block , 2011, Numer. Linear Algebra Appl..

[23]  Jan Mandel,et al.  On block diagonal and Schur complement preconditioning , 1990 .

[24]  Andrew J. Wathen,et al.  Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations , 2002, Numerische Mathematik.

[25]  John Ambrosiano,et al.  A finite element formulation of the Darwin PIC model for use on unstructured grids , 1995 .

[26]  C. W. Nielson,et al.  A multidimensional quasineutral plasma simulation model , 1978 .

[27]  Zhiming Chen,et al.  Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients , 2000, SIAM J. Numer. Anal..

[28]  Jun Zou,et al.  Finite element convergence for the Darwin model to Maxwell's equations , 1997 .

[29]  Michael K. Ng,et al.  On Inexact Preconditioners for Nonsymmetric Matrices , 2005, SIAM J. Sci. Comput..

[30]  Xiaoming Yuan,et al.  Local Linear Convergence of the Alternating Direction Method of Multipliers for Quadratic Programs , 2013, SIAM J. Numer. Anal..

[31]  Fang Chen,et al.  Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices , 2013, Numerical Algorithms.

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

[33]  Nicholas I. M. Gould,et al.  CUTEr and SifDec: A constrained and unconstrained testing environment, revisited , 2003, TOMS.

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

[35]  R. Lehoucq,et al.  A Primal-Based Penalty Preconditioner for Elliptic Saddle Point Systems , 2006, SIAM J. Numer. Anal..

[36]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[37]  Zhen Wang,et al.  A Relaxed Dimensional Factorization preconditioner for the incompressible Navier-Stokes equations , 2011, J. Comput. Phys..

[38]  Piotr Krzyzanowski,et al.  On Block Preconditioners for Nonsymmetric Saddle Point Problems , 2001, SIAM J. Sci. Comput..

[39]  H. Elman,et al.  Efficient preconditioning of the linearized NavierStokes equations for incompressible flow , 2001 .

[40]  Naimin Zhang,et al.  Constraint preconditioners for solving singular saddle point problems , 2013, J. Comput. Appl. Math..

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