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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.

We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.

SUMMARYA mass-conserving Level-Set method to model bubbly ows is presented. The method can handle high density-ratio ows with complex interface topologies, such as ows with simultaneous occurrence of bubbles and droplets. Aspects taken into account are: a sharp front (density changes abruptly), arbitrarily shaped interfaces, surface tension, buoyancy and coalescence of droplets=bubbles. Attention is paid to mass-conservation and integrity of the interface.The proposed computational method is a Level-Set method, where a Volume-of-Fluid function is used to conserve mass when the interface is advected. The aim of the method is to combine the advantages of the Level-Set and Volume-of-Fluid methods without the disadvantages. The ow is computed with a pressure correction method with the Marker-and-Cell scheme. Interface conditions are satisÿed by means of the continuous surface force methodology and the jump in the density ÿeld is maintained similar to the ghost uid method for incompressible ows.