The subject of this paper is the application of the multi-grid method to the solution of -V. (D(x, y)VU(x, y))+(x, y)U(x, y)=f (x, y) in a bounded region f of R where D is positive and D, tr, and f are allowed to be discontinuous across internal boundaries F of fL The emphasis is on discontinuities of orders of magnitude in D, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of D VU across F. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.

No abstract

No abstract

It has been shown that the usual Galerkin procedure applied to first order hyperbolic equations does not yield the optimum L rate of convergence for all admissible finite-dimensional subspaces. Two methods are presented which overcome this difficulty.

No abstract

No abstract

No abstract

The purpose of this paper is to report on a preliminary study of a new multigrid scheme for solving three-dimensional problems of the type that arise from pressure equations in reservoir simulation. The basic idea of this new approach is to use both semi-coarsening and plane relaxation in order to accommodate all types of anisotropies and heterogeneities. The scheme also simplifies and reduces the cost of the setup involved in automatic coarsening of the fine-grid equations. This paper presents the results of the new approach applied to several known test problems. A comparison of performance and complexity is made with the best solvers now in widespread use in reservoir simulation.