Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

Abstract

<p style='text-indent:20px;'>In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water flow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy <inline-formula><tex-math id="M1">\begin{document}$ H = O(\sqrt{h}) $\end{document}</tex-math></inline-formula>.</p>