@Article{AAMM-13-1418, author = {Bian , HuanyingShen , Yedan and Hu , Guanghui}, title = {An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {6}, pages = {1418--1440}, abstract = {

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0218}, url = {http://global-sci.org/intro/article_detail/aamm/19429.html} }