scholarly journals Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations

2015 ◽  
Vol 85 (298) ◽  
pp. 549-580 ◽  
Author(s):  
Clément Cancès ◽  
Cindy Guichard
Keyword(s):  
Finite Element ◽  
Parabolic Equations ◽  
Control Volume ◽  
Degenerate Parabolic Equations ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Degenerate Parabolic ◽  
Control Volume Finite Element

scholarly journals Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy

2018 ◽  
Vol 52 (4) ◽  
pp. 1533-1567 ◽  
Author(s):  
Ahmed Ait Hammou Oulhaj ◽  
Clément Cancès ◽  
Claire Chainais–Hillairet
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Stability Property ◽  
Control Volume ◽  
Positive Control ◽  
Richards Equation ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Anisotropy Tensor ◽  
Control Volume Finite Element

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.

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