Nonlinear effects on the convergence of Picard iterations for the solution of Gauss-quadrature based finite element approximations of Richards' equation

摘要

Unsaturated flow problems are usually solved by means of numerical approximations of the nonlinear Richards equation. While space-centered finite differences or lumped mass Galerkin approaches are common methods for such numerical approximations, Gauss Quadrature based finite element approximations have a number of advantages and therefore are also regularly used to simulate unsaturated flow. In previous papers, the effect of nonlinearities in the stability of θ/finite difference or θ/lumped finite element approximation, as well as the convergence properties of both Newton and Picard Iterations applied to such numerical solutions have been studied. In those studies, Frechet-Taylor expansions of discrete operators and iteration errors and localization approaches were used and numerical experiments confirmed the theoretical results. In this paper, the same concepts and methodologies are applied to the analysis of convergence of Gauss Quadrature finite element approximations of Richards' equation using Picard iterations. Numerical experiments confirming the theoretical results are also presented.