PCG solutions of flow problems in random porous media using mixed finite elements

Abstract

The behavior of preconditioned conjugate gradient (PCG) solutions of flow problems in random porous media is examined. The flow problems are discretized with the mixed finite element method, the resulting matrix is reduced in size by block Gauss elimination, and PCG is applied to the resulting problem. As a result of the matrix reduction and the need to store the resultant matrix implicitly because of storage limitations for large problems, the selection of a preconditioner is not obvious. Four heuristic preconditioners, selected because of the simplicity with which they can be constructed from the implicit matrix representation and because of their potential for high performance computers, are tested: diagonal, tridiagonal, scaled diagonal, and scaled tridiagonal. Tests on fields with random hydraulic conductivity, with both isotropic and anisotropic correlation functions and with a large range of standard deviations, are performed. The scaled tridiagonal and diagonal preconditioners prove very effective and efficient for a broad range of problems, even for conductivity variations of 25 orders of magnitude.