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A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm. Comparisons have been made with other well-known methods. In all test problems the new combination was faster than the other methods.

Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a time-dependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion is not valid in this problem. After diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number 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 approximation of the eigenvectors is proposed.

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To predict the presence of oil and natural gas in a reservoir, it is important to know the fluid pressure in the rock formations. A mathematical model for the prediction of the fluid pressure history is given by a time-dependent diffusion equation. Application of the finite-element method leads to systems of linear equations. A complication is that the underground consists of layers with very large contrasts in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the ICCG method has been observed, and a classical termination criterion is not valid in this problem. In [19] we have shown that the number of small eigenvalues of the diagonally scaled matrix is equal to the number of high-permeability domains, which are not connected to a Dirichlet boundary. In this paper the proof is extended to an Incomplete Cholesky decomposition. To annihilate the bad effect of these small eigenvalues on the convergence, the Deflated ICCG method is used. In [19] we have shown how to construct a deflation subspace for the case of a set of more or less parallel layers. That subspace proved to be a good approximation of the span of the "small" eigenvectors. As a result of this, the convergence of DICCG is independent of the contrasts in the permeabilities. In this paper it is shown how to construct deflation vectors even in the case of very irregular shaped layers, and layers with so-called inclusions. A theoretical investigation and numerical experiments show that the DICCG method is not sensitive to small perturbations of the deflation vectors. The efficiency of the DICCG method is illustrated by numerical experiments.