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CivilComp Proceedings
ISSN 17593433 CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis and B.H.V. Topping
Paper 75
QuasiNewton Preconditioners for the Solution of Large Nonlinear Systems in Porous Media L. Bergamaschi^{1}, R. Bru^{2}, A. Martínez^{1} and M. Putti^{1}
^{1}Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy
, "QuasiNewton Preconditioners for the Solution of Large Nonlinear Systems in Porous Media", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 75, 2008. doi:10.4203/ccp.89.75
Keywords: quasiNewton method, Krylov iterations, updating preconditioners, inexact Newton method, two phase flow, porous media.
Summary
Newton's method for the solution of systems of nonlinear equations
requires the solution of a number of linear systems with the Jacobian J as the coefficient matrix. When J is large and sparse, for example for problems
arising from the discretization of a nonlinear PDE, which is our case as we describe below, the preconditioned Krylov based iterative schemes can be employed for the
solution of the linear system, so that two nested iterative
procedures need to be implemented [2,3]. A crucial issue, for the reduction of the total linear
iterations, is to use efficient preconditioning techniques.
In general, ILUtype preconditioners can be employed
and calculated at each nonlinear iteration. In this case, a large cost of the calculation of the preconditioners has been paid.
In this work preconditioners for solving the linear systems of the Newton method in each nonlinear iteration are studied. The preconditioner is defined by means of a Broydentype rankone update of a given initial preconditioner, at each nonlinear iteration. Our sequence of preconditioners built like this, are bounded in the sense that the norm of the matrix obtained from the identity matrix minus the preconditioner times the Jacobian, in each outer iteration [1]. The approach proposed in this paper is to solve the inner systems of the Newton method with an iterative Krylov subspace method, starting with the ILU(0) preconditioner, computed from the initial Jacobian, and to update this preconditioner using a rank one sum. A sequence of preconditioners P_{k} can thus be defined by imposing the secant condition, as used in the implementation of quasiNewton methods. With this approach an algorithm has been constructed where the preconditioners are updated until a k_{max} number of nonlinear iterations. Afterwards a new ILU(0) preconditioner is computed as a restarted procedure. We have applied the obtained algorithm to the solution of the nonlinear system of algebraic equations arising from the discretization of the highly nonlinear equations governing the two phase model in porous media [4,5]. The discretization has been done using linear finite elements (triangles in two and tetrahedra in three dimensions) yielding a system of first order differential equations integrated in time via backward Euler forward difference method [6]. The numerical results show an improvement both in terms of iteration number and CPU time with respect to the ILU(0) preconditioner computed in each nonlinear iteration. In addition these improvements are mainly when the values of the k_{max} parameter are small.
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