RMCP: Relaxed Mixed Constraint Preconditioners for Saddle Point Linear Systems arising in Geomechanics

Abstract A major computational issue in the Finite Element (FE) integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system. Because of ill-conditioning, the iterative solution, which is recommended in large size 3D settings, requires the computation of a suitable preconditioner to guarantee convergence. In this paper the coupled system is solved by a Krylov subspace method preconditioned by a Relaxed Mixed Constraint Preconditioner (RMCP) which is a generalization based on a parameter ω of the Mixed Constraint Preconditioner (MCP) developed in [7] . Choice of optimal ω is driven by the spectral distribution of suitable symmetric positive definite (SPD) matrices. Numerical tests performed on realistic 3D problems reveal that RMCP accelerates Krylov subspace solvers by a factor up to three with respect to MCP.

[1]  Marco Vianello,et al.  A massively parallel exponential integrator for advection-diffusion models , 2009, J. Comput. Appl. Math..

[2]  Ilaria Perugia,et al.  Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 2000, Numer. Linear Algebra Appl..

[3]  Giuseppe Gambolati,et al.  Ill-conditioning of finite element poroelasticity equations , 2001 .

[4]  Luca Bergamaschi,et al.  An Efficient Parallel MLPG Method for Poroelastic Models , 2009 .

[5]  Luca Bergamaschi,et al.  Performance and robustness of block constraint preconditioners in finite element coupled consolidation problems , 2009 .

[6]  Luca Bergamaschi,et al.  FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics , 2011, J. Comput. Appl. Math..

[7]  Jan Vlcek,et al.  Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems , 1998, Numer. Linear Algebra Appl..

[8]  Luca Bergamaschi,et al.  Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations , 2008, J. Comput. Phys..

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

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

[11]  L. Luksan,et al.  Indefinitely preconditioned inexact Newton method for large sparse equality constrained non‐linear programming problems , 1998 .

[13]  Luca Bergamaschi,et al.  Preconditioning Indefinite Systems in Interior Point Methods for Optimization , 2004, Comput. Optim. Appl..

[14]  Luca Bergamaschi,et al.  Parallel Inexact Constraint Preconditioners for Saddle Point Problems , 2011, Euro-Par.

[15]  M. Putti,et al.  Numerical comparison of iterative eigensolvers for large sparse symmetric positive definite matrices , 2002 .

[16]  Domenico Baù,et al.  Basin-scale compressibility of the northern Adriatic by the radioactive marker technique , 2002 .

[17]  L. Kolotilina,et al.  Factorized Sparse Approximate Inverse Preconditionings I. Theory , 1993, SIAM J. Matrix Anal. Appl..

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

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

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

[21]  G. Gambolati,et al.  Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem , 1997 .

[22]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[23]  Luca Bergamaschi,et al.  Parallel Acceleration of Krylov Solvers by Factorized Approximate Inverse Preconditioners , 2004, VECPAR.

[24]  M. Benzi,et al.  A comparative study of sparse approximate inverse preconditioners , 1999 .

[25]  Michele Benzi,et al.  Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method , 2000, SIAM J. Sci. Comput..

[26]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[27]  Lily Yu. Kolotilina,et al.  Factorized sparse approximate inverse preconditionings. IV: Simple approaches to rising efficiency , 1999, Numer. Linear Algebra Appl..

[28]  Luca Bergamaschi,et al.  Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations , 2007 .

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