The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical properties of new schemes, by developing a generic process for eliminating unknowns via barycentric condensation, and by designing a concept of discrete functional analysis toolbox for schemes based on polytopal meshes.
DOI : 10.1051/m2an/2015079
Mots-clés : Gradient scheme, gradient discretisation, numerical scheme, diffusion equations, convergence analysis, discrete functional analysis
@article{M2AN_2016__50_3_749_0,
author = {Droniou, J\'erome and Eymard, Robert and Herbin, Rapha\`ele},
title = {Gradient schemes: {Generic} tools for the numerical analysis of diffusion equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {749--781},
publisher = {EDP-Sciences},
volume = {50},
number = {3},
year = {2016},
doi = {10.1051/m2an/2015079},
zbl = {1346.65042},
mrnumber = {3507272},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015079/}
}
TY - JOUR AU - Droniou, Jérome AU - Eymard, Robert AU - Herbin, Raphaèle TI - Gradient schemes: Generic tools for the numerical analysis of diffusion equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 749 EP - 781 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015079/ DO - 10.1051/m2an/2015079 LA - en ID - M2AN_2016__50_3_749_0 ER -
%0 Journal Article %A Droniou, Jérome %A Eymard, Robert %A Herbin, Raphaèle %T Gradient schemes: Generic tools for the numerical analysis of diffusion equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 749-781 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015079/ %R 10.1051/m2an/2015079 %G en %F M2AN_2016__50_3_749_0
Droniou, Jérome; Eymard, Robert; Herbin, Raphaèle. Gradient schemes: Generic tools for the numerical analysis of diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 749-781. doi : 10.1051/m2an/2015079. http://www.numdam.org/articles/10.1051/m2an/2015079/
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