Quasi-Newton acceleration of ILU preconditioners for nonlinear two-phase flow equations in porous media

In this work, preconditioners for the iterative solution by Krylov methods of the linear systems arising at each Newton iteration are studied. The preconditioner is defined by means of a Broyden-type rank-one update of a given initial preconditioner, at each nonlinear iteration, as described in [5] where convergence properties of the scheme are theoretically proved. This acceleration is employed in the solution of the nonlinear system of algebraic equations arising from the Finite Element discretization of two-phase flow model in porous media. We report numerical results of the application of this approach when the initial preconditioner is chosen to be the incomplete LU decomposition of the Jacobian matrix at the initial nonlinear stage. It is shown that the proposed acceleration reduces the number of linear iterations needed to achieve convergence. Also, the cost of computing the preconditioner is reduced as this operation is made only once at the beginning of the Newton iteration.