We extend the nonlinear Control Volume Finite Element scheme of [C. Cancès and C. Guichard, Math. Comput. 85 (2016) 549–580]. to the discretization of Richards equation. This scheme ensures the preservation of the physical bounds without any restriction on the mesh and on the anisotropy tensor. Moreover, it does not require the introduction of the so-called Kirchhoff transform in its definition. It also provides a control on the capillary energy. Based on this nonlinear stability property, we show that the scheme converges towards the unique solution to Richards equation when the discretization parameters tend to 0. Finally we present some numerical experiments to illustrate the behavior of the method.
Mots-clés : Unsaturated porous media flow, Richards equation, nonlinear discretization, nonlinear stability, convergence analysis
@article{M2AN_2018__52_4_1533_0,
author = {Ait Hammou Oulhaj, Ahmed and Canc\`es, Cl\'ement and Chainais{\textendash}Hillairet, Claire},
title = {Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for {Richards} equation with anisotropy},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1533--1567},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017012},
zbl = {1407.65160},
mrnumber = {3875296},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017012/}
}
TY - JOUR AU - Ait Hammou Oulhaj, Ahmed AU - Cancès, Clément AU - Chainais–Hillairet, Claire TI - Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1533 EP - 1567 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017012/ DO - 10.1051/m2an/2017012 LA - en ID - M2AN_2018__52_4_1533_0 ER -
%0 Journal Article %A Ait Hammou Oulhaj, Ahmed %A Cancès, Clément %A Chainais–Hillairet, Claire %T Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1533-1567 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017012/ %R 10.1051/m2an/2017012 %G en %F M2AN_2018__52_4_1533_0
Ait Hammou Oulhaj, Ahmed; Cancès, Clément; Chainais–Hillairet, Claire. Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1533-1567. doi : 10.1051/m2an/2017012. http://www.numdam.org/articles/10.1051/m2an/2017012/
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