An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients

Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressu res is given by a time-dependent diffusion equation. Application of the fin ite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in per meability. This implies that the symmetric and positive definite coefficien t matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion i s not valid in this problem. After diagonal scaling of the matrix the numbe r of extreme eigenvalues is reduced and it is proved to be equal to the num ber of layers with a high permeability, For the IC preconditioner the same behavior is observed. To annihilate the effect of the extreme eigenvalues a deflated CG method is used, The convergence rate improves considerably and the termination criterion becomes again reliable. Finally a cheap approxim ation of the eigenvectors is proposed. (C) 1999 Academic Press.