Keywords: parallel computing, preconditioning, Krylov subspace methods, coupled consolidation.

Large and sparse linear systems of saddle point type arise in many engineering applications such as mixed finite element discretizations of PDEs, constrained optimization and coupled consolidation problems. Iterative solution is recommended as opposed to direct factorization methods because of the extremely large size of these systems. However, well established iterative methods such as Krylov subspace methods are very slow or even fail to converge if not conveniently preconditioned.

In this paper we propose and describe a parallel implementation of a block preconditioner in the solution of saddle point linear systems arising from finite element (FE) discretization of three-dimensional coupled consolidation problems. The constraint preconditioners for Krylov solvers in the solution of saddle point problems have been studied by a number of authors. Bergamaschi et al. [1,2] have developed a mixed constraint preconditioner (MCP) which couples two explicit-implicit approximations of the inverse of the (1,1) block (K in the sequel) provided by an approximate inverse preconditioner such as AINV, and the IC preconditioner, respectively.

In this paper we propose a parallel preconditioner (FSAI-MPC) which uses two factorized sparse approximate inverse (FSAI) [3] approximations, G1 and G2, of the inverse of K. The FSAI preconditioner has been successfully implemented in parallel and tested on large geomechanical models [4]. The sparser approximation G2 is used to explicitly construct the Schur complement matrix which in its turn is preconditioned by another FSAI approximation.

We have developed a fully parallel code which implements the BiCGSTAB solver accelerated with the parallel MCP preconditioner described above. The code is written in Fortran 90 and uses the MPI standard to perform interprocessor communication. We present numerical results of several run solving a number of three-dimensional test cases of large size arising from three-dimensional finite element discretization of coupled consolidation problems. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. The numerical results of a number of test cases of size up to 2 million unknowns and 120 million nonzeros show the perfect scalability of the overall code up to 256 processors. The proposed iterative method is shown to outperform the well-established pARMS multilevel solver on the largest problem.

1

L. Bergamaschi, M. Ferronato, G. Gambolati, "Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations", Comp. Methods App. Mech. Engrg., 196, 2647-2656, 2007. doi:10.1016/j.cma.2007.01.013

2

L. Bergamaschi, M. Ferronato, G. Gambolati, "Mixed constraint preconditioners for the solution to FE coupled consolidation equations", J. Comp. Phys., 227, 9885-9897, 2008. doi:10.1016/j.jcp.2008.08.002

3

L. Yu. Kolotilina, A.A. Nikishin, A.Yu. Yeremin, "Factorized sparse approximate inverse preconditionings IV. Simple approaches to rising efficiency", Numer. Lin. Alg. Appl., 6, 515-531, 1999. doi:10.1002/(SICI)1099-1506(199910/11)6:7<515::AID-NLA176>3.0.CO;2-0

4

L. Bergamaschi, A. Martinez, G. Pini, "An efficient parallel MLPG method for poroelastic models", CMES: Computer and Modeling in Engineering & Sciences, 49(3), 191-216, 2009. doi:10.3970/cmes.2009.049.191

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