TY - JOUR
T1 - An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers
AU - Bian , Huanying
AU - Shen , Yedan
AU - Hu , Guanghui
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1418
EP - 1440
PY - 2021
DA - 2021/08
SN - 13
DO - http://doi.org/10.4208/aamm.OA-2020-0218
UR - https://global-sci.org/intro/article_detail/aamm/19429.html
KW - Non-equilibrium Richard equation, $h$-adaptive mesh method, a posteriori error estimation, fingering phenomenon, porous media flow.
AB - <p style="text-align: justify;">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&nbsp;mechanism on generating fingers is discussed by numerically studying several examples.</p>