Gradient discretisation method information

Exact solution
${\overline {u}}(x)={\frac {3}{4}}\left({0.5}^{4/3}-|x-0.5|^{4/3}\right)$
of the p-Laplace problem $-(|{\overline {u}}'|^{2}{\overline {u}}')'(x)=1$ on the domain [0,1] with ${\overline {u}}(0)={\overline {u}}(1)=0$ (black line) and approximate one (blue line) computed with the first degree discontinuous Galerkin method plugged into the GDM (uniform mesh with 6 elements).

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 ^[1] (the quantities $C_{D}$ , $S_{D}$ and $W_{D}$ , 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.^[2] 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.^[3]

Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements, Mixed Finite Elements, nonconforming Finite Elements, and, in the case of more recent schemes, the Discontinuous Galerkin method, Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes

^ R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.
^ 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.
^ 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.

[1] R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.

[droniou-2] 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.

[3] 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.

Gradient discretisation method information

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Gradient discretisation method

List of numerical analysis topics

GDM

Material point method

Boundary element method

Verlet integration

Schwarz alternating method

Computational fluid dynamics

Flux limiter

Normal distributions transform

MUSCL scheme