Computational orders of convergence of iterative methods for Richards' equation

TL;DR

The paper tackles the computational assessment of iterative solvers for Richards' equation in variably saturated porous media by framing convergence through computational orders of convergence. It systematically analyzes Newton's method and the $L$-scheme within an implicit FEM and an explicit FDM, including the role of Anderson Acceleration, on a 2D test problem. Key findings show that Anderson Acceleration markedly improves the FEM performance by halving iterations and doubling speed for the $L$-scheme, while its effect on the explicit FDM is negligible or detrimental; moreover, the explicit FDM $L$-scheme can be faster overall than the FEM with AA despite more iterations. The work demonstrates how convergence-order analysis can quantify and compare solver efficiency for nonlinear Richards' equation and informs practical solver choices in porous media simulations.

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.

Figures (12)

• Figure 1: FEM: Norms of successive corrections provided by Newton's method.
• Figure 2: FEM: Estimation of $Q$-order $p$ for the convergence of Newton's method.
• Figure 3: FEM: Estimation of $R$-order 2 for Newton's method.
• Figure 4: FEM: Estimation of the quotient $Q_2$ from definition (\ref{['eq:Qp']}) for Newton's method.
• Figure 5: FEM: Estimation of the quotient $Q_1$ from definition (\ref{['eq:Qp']}) for $L$-scheme ($L=0.15$).