Geodesic
Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy
Ait Hammou Oulhaj, Ahmed1  ; Cancès, Clément1  ; Chainais–Hillairet, Claire2 
1 Univ. Lille, CNRS, UMR 8524, Inria — Laboratoire Paul Painlevé, 59000 Lille, France
2 Univ. Lille, CNRS, UMR 8524, Inria — Laboratoire Paul Painlevé, 59000 Lille, France
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1533-1567

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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.

DOI : 10.1051/m2an/2017012
Classification : 65M12, 65M08, 76S05
Keywords: Unsaturated porous media flow, Richards equation, nonlinear discretization, nonlinear stability, convergence analysis

Ait Hammou Oulhaj, Ahmed 1 ; Cancès, Clément 1 ; Chainais–Hillairet, Claire 2

1 Univ. Lille, CNRS, UMR 8524, Inria — Laboratoire Paul Painlevé, 59000 Lille, France
2 Univ. Lille, CNRS, UMR 8524, Inria — Laboratoire Paul Painlevé, 59000 Lille, France 
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     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 },
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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

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