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.

翻译：部分饱和多孔介质中流动的数值解面临控制方程Richards方程的非线性和椭圆-抛物线退化带来的挑战。因此需要采用迭代方法来处理流动问题的复杂性。迭代过程中逐次修正的范数形成正数序列。利用计算收敛阶的定义及抽象收敛序列的理论结果，可以评估和比较不同的迭代方法。在此框架下，我们分析了隐式有限元法（FEM）的牛顿法和$L$格式方法，以及显式有限差分法（FDM）的$L$格式方法。我们还研究了安德森加速（AA）对隐式和显式$L$格式方法的影响。通过二维测试问题，我们发现AA使迭代次数减半，并使FEM格式的收敛速度提高两倍。对于FDM方法，AA并未减少迭代次数，反而增加了计算量。相反，作为显式方法，未采用AA的FDM的$L$格式方法比采用AA的FEM的$L$格式方法更快且同样精确。