A simplified Newton Iteration Method with linear finite elements for transient unsaturated flow

Abstract

A simplified Newton iteration method with Galerkin linear finite elements is developed for the solution of the nonlinear transient unsaturated flow equation in “mixed” form. The use of the Galerkin linear finite element method in the spatial discretization produces a recurrence equation in which the interaode hydraulic conductivity is represented by an integrated average expression which generates the exact Darcy flow flux through each element under the steady horizontal flow condition. The numerical computation of the integrated average hydraulic conductivity using Simpson's 1/3 rule leads to an accurate solution even in the presence of a sharp front. In the time discretization using a backward Euler scheme, the standard Newton iteration method is simplified by taking account of the integrated average expression of the hydraulic conductivity and the diffusion‐dominated nature of the flow process. The resulting two‐level scheme is of the same efficiency on a per‐iteration basis as the modified Picard scheme but converges at a faster rate.