Gradient discretisation method: Difference between revisions

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{{Differential equations}}

In numerical mathematics, the '''gradient discretisation method''' ('''GDM)''') is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless).

Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM <ref>'''R. Eymard, C. Guichard, and R. Herbin.''' Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.</ref> (the quantities <math>C_{D}</math>, <math>S_{D}</math> and <math>W_{D}</math>, [[#The example of a linear diffusion problem|see below]]). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data.<ref name=droniou>'''J. Droniou, R. Eymard, T. Gallouët, and R. Herbin.''' Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS), 23(13):2395–2432, 2013.</ref> [[#Some non-linear problems with complete convergence proofs of the GDM|Non-linear models]] for which such convergence proof of the GDM have been carried out comprise: the [[Stefan problem]] which is modelling a melting material, two-phase flows in porous media, the [[Richards equation]] of underground water flow, the fully non-linear Leray—Lions equations.<ref>'''J. Leray and J. Lions.''' Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France, 93:97–107, 1965.</ref>