Preconditioners for regularized saddle point problems with an application for heterogeneous Darcy flow problems

Saddle point problems arise in the modeling of many important practical situations. Preconditioners for the corresponding matrices of block-triangular form, based on coupled inner-outer iteration methods, are analyzed and applied to a Darcy flow problem, possibly with strong heterogeneity and non-symmetric saddle point systems. Using proper regularized forms of the given matrix and its preconditioner, it is shown that the eigenvalues cluster about one or two points on the real axis for large values of the regularization parameters and that eigenvalue bounds do not depend on this variation. Therefore, just two outer iterations can suffice. On the other hand, several iterations may be needed to solve the inner iteration systems.

[1]  Yalchin Efendiev,et al.  Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces , 2010, Multiscale Model. Simul..

[2]  P. Vassilevski,et al.  Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates , 1996 .

[3]  Mary F. Wheeler,et al.  Efficient algorithms for multiscale modeling in porous media , 2010, Numer. Linear Algebra Appl..

[4]  Owe Axelsson,et al.  Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices , 2012, Comput. Vis. Sci..

[5]  Owe Axelsson,et al.  Preconditioning of Boundary Value Problems Using Elementwise Schur Complements , 2009, SIAM J. Matrix Anal. Appl..

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

[7]  Zhi-Hao Cao.  A note on spectrum distribution of constraint preconditioned generalized saddle point matrices , 2009, Numer. Linear Algebra Appl..

[8]  Valeria Simoncini,et al.  Block triangular preconditioners for symmetric saddle-point problems , 2004 .

[9]  P. Vassilevski.  Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations , 2008 .

[10]  Owe Axelsson,et al.  Preconditioning methods for linear systems arising in constrained optimization problems , 2003, Numer. Linear Algebra Appl..

[11]  Eun-Jae Park,et al.  Mixed finite element methods for nonlinear second-order elliptic problems , 1995 .

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

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

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

[15]  Panayot S. Vassilevski,et al.  Preconditioning Mixed Finite Element Saddle‐point Elliptic Problems , 1996 .

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

[17]  O. Axelsson,et al.  A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning , 1991 .

[18]  Alexandre Ern,et al.  A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection–diffusion equations , 2007 .

[19]  EL L..  A priori and a posteriori analysis of nonconforming finite elements with face penalty for advection – diffusion equations , .

[20]  Owe Axelsson,et al.  Preconditioning of matrices partitioned in 2 × 2 block form: eigenvalue estimates and Schwarz DD for mixed FEM , 2010, Numer. Linear Algebra Appl..

[21]  G. F. Carey,et al.  Least-Squares Mixed Finite Element Methods for Non-Selfadjoint Elliptic Problems: II. Performance of Block-ILU Factorization Methods , 1995, SIAM J. Sci. Comput..

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

[23]  Johannes Kraus,et al.  Additive Schur Complement Approximation and Application to Multilevel Preconditioning , 2012, SIAM J. Sci. Comput..

[24]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[25]  Owe Axelsson,et al.  A general approach to analyse preconditioners for two-by-two block matrices , 2013, Numer. Linear Algebra Appl..

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

[27]  Axel Klawonn,et al.  Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term , 1998, SIAM J. Sci. Comput..

[28]  Walter Zulehner,et al.  Analysis of iterative methods for saddle point problems: a unified approach , 2002, Math. Comput..

[29]  Uwe Fink.  Finite Element Solution Of Boundary Value Problems Theory And Computation , 2016 .

[30]  D. Schötzau,et al.  Preconditioners for saddle point linear systems with highly singular blocks. , 2006 .

[31]  Jaroslav Haslinger,et al.  Projected Schur complement method for solving non‐symmetric systems arising from a smooth fictitious domain approach , 2007, Numer. Linear Algebra Appl..

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

[33]  O. Axelsson.  Preconditioning of Indefinite Problems by Regularization , 1979 .

[34]  Catherine Elizabeth Powell,et al.  Optimal Preconditioning for Raviart-Thomas Mixed Formulation of Second-Order Elliptic Problems , 2003, SIAM J. Matrix Anal. Appl..

[35]  Ivan Yotov,et al.  Coupling Fluid Flow with Porous Media Flow , 2002, SIAM J. Numer. Anal..