标题： 迭代方法求解Richards方程的收敛阶数 显示英文标题

标题： Computational orders of convergence of iterative methods for Richards' equation

摘要： 对于部分饱和多孔介质中的流动问题的数值解，涉及控制Richards方程的非线性和椭圆-抛物退化性，这带来了挑战。因此，需要迭代方法来处理流动问题的复杂性。迭代过程中连续修正的范数形成正数序列。可以利用计算收敛阶数的定义和抽象收敛序列的理论结果来评估和比较不同的迭代方法。我们在这一框架下分析了隐式有限元方法（FEM）的牛顿法和$L$-方案，以及显式有限差分方法（FDM）的$L$-方案。我们还研究了Anderson加速（AA）对隐式和显式$L$-方案的影响。考虑到一个二维测试问题，我们发现AA将迭代次数减少了一半，并使FEM方案的收敛速度提高了一倍。而对于FDM方法，AA并没有减少迭代次数，反而增加了计算量。相反，没有AA的FDM$L$-方案由于其显式性质，比带有AA的FEM$L$-方案更快且同样准确。 显示英文摘要

摘要： 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.