A numerical investigation of the conjugate gradient method as applied to three‐dimensional groundwater flow problems in randomly heterogeneous porous media

Field‐scale modeling of three‐dimensional groundwater flow in randomly heterogeneous porous media is considered. The system of linear equations resulting from the finite difference discretization of this problem may easily involve more than a million unknowns. Problems of such magnitude can only be solved practically on supercomputers and require an efficient iterative solution method that is well suited to the particular computer architecture being used. The preconditioned conjugate gradient method is highly efficient for this groundwater flow problem, and by using an appropriate preconditioning matrix the method can be adapted to different computers. The numerical software package CgCode enhances this adaptability; the user of CgCode must only provide subroutines for preconditioning and matrix‐vector multiplication. These subroutines should be tailored to the specific problem and machine architecture. Numerical results demonstrating the efficiency of polynomial preconditioning for the conjugate gradient method on both a vector machine (Cray X‐MP/48) and a vector‐parallel machine (Alliant FX/8) are presented.