Keywords: finite element method, saddle point problem, Krylov iterative methods, preconditioners, incomplete factorizations, contact problems.

The finite element modelling of contact problems is based on the minimization of the total potential energy functional under displacement constraints. Lagrange multiplier method can be used to solve this minimization problem. In this case, the unknowns of the linear system are the displacements and the contact stresses. The coefficient matrix of this system is symmetric, indefinite and ill-conditioned, and so this is a saddle point problem. Moreover, this matrix typically is large and sparse and the system must be solved using iterative methods, usually Krylov algorithms. There is a wide bibliography related with the saddle point solvers [1].

To solve this linear systems we present a mixed constraint preconditioner similar to the preconditioner introduced by Bergamaschi et al. in [2], using a balanced incomplete factorization (BIF) described in [3] of the block (1,1) of the coefficient matrix. The advantage of this factorization is that approximations for the Choleski factors of the matrix and of its inverse are computed at the same time.

This preconditioner has been used to solve contact problems obtained from fretting-fatigue analysis. We also include some numerical results to compare the performance of this preconditioner with others already known in the literature, such as block diagonal preconditioners using incomplete Cholesky factorizations. Since there are large differences in the magnitude of the entries in the matrices considered in the experiments, a diagonal scaling has been applied such that all entries vary between -1 and 1. Moreover, because of the block (1,1) is close to a singular matrix, we have also applied the augmented Lagrangian technique [4]. In the three examples considered, the mixed constraint preconditioner using BIF algorithm provided the best results.

1

M. Benzi, G.H. Golub, J. Liesen, "Numerical Solution of Saddle Point Problems", Acta Numerica, 14, 1-137 (invited survey paper), 2005. doi:10.1017/S0962492904000212

2

L. Bergamaschi, M. Ferronato, G. Gambolati, "Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations", Journal of Computational Physics, 227, 9885-9897, 2008. doi:10.1016/j.jcp.2008.08.002

3

R. Bru, J. Marín, J. Mas, M. Tuma, "Balanced incomplete factorization", SIAM J. Sci. Comput., 30 (5), 2302-2318, 2007. doi:10.1137/070696088

4

M. Fortin, R. Glowinski, "Augmented Lagrangian Methods: Application to the Solution of Boundary-Value Problems", Stud. Math. Appl., 15, North-Holland, Amsterdam, 1983.

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