Computational orders of convergence of iterative methods for Richards' equation

TL;DR

This paper evaluates the convergence behavior of iterative methods for solving Richards' equation, analyzing the impact of Anderson Acceleration on both implicit and explicit schemes through theoretical and numerical comparisons.

Contribution

It introduces a framework for analyzing the orders of convergence of iterative methods applied to Richards' equation, including the effects of Anderson Acceleration on different schemes.

Findings

Anderson Acceleration halves iterations for FEM scheme

AA speeds up convergence of implicit FEM by two times

AA does not reduce iterations for explicit FDM, may increase effort

Abstract

Numerical solutions for flows in partially saturated porous media pose challenges related to the non-linearity and elliptic-parabolic degeneracy of the governing Richards' equation. Iterative methods are therefore required to manage the complexity of the flow problem. Norms of successive corrections in the iterative procedure form sequences of positive numbers. Definitions of computational orders of convergence and theoretical results for abstract convergent sequences can thus be used to evaluate and compare different iterative methods. We analyze in this frame Newton's and $L$-scheme methods for an implicit finite element method (FEM) and the $L$-scheme for an explicit finite difference method (FDM). We also investigate the effect of the Anderson Acceleration (AA) on both the implicit and the explicit $L$-schemes. Considering a two-dimensional test problem, we found that the AA halves the number of iterations and renders the convergence of the FEM scheme two times faster. As for the FDM approach, AA does not reduce the number of iterations and even increases the computational effort. Instead, being explicit, the FDM $L$-scheme without AA is faster and as accurate as the FEM $L$-scheme with AA.