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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 (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. [[#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 model of a melting material, two-phase flows in porous media, the Richards equation of underground water flow, the fully non-linear Leray—Lions equations [Leray ''et al'', 1965].
Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements,
▲Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements, Raviart—Thomas Mixed Finite Elements, the <math>P^1</math> non-conforming Finite Elements, and, in the case of more recent schemes, the Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes.
==The example of a linear diffusion problem==
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:<math>\quad (1) \qquad \qquad -\Delta \overline{u} = f,</math>
where <math>f\in L^2(\Omega)</math>. The usual sense of weak solution [Brezis, 2011] to this model is:
:<math>\quad (2) \qquad\mbox{Find }\overline{u}\in H^1_0(\Omega)\mbox{ such that, for all } \overline{v} \in H^1_0(\Omega),\quad \int_{\Omega} \nabla \overline{u}(x)\cdot\nabla \overline{v}(x) dx = \int_{\Omega} f(x)\overline{v}(x) dx. </math>
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The "mass-lumped" <math>P^1</math> finite element case enters the framework of the GDM, replacing <math>\Pi_D u</math> by <math>\widetilde{\Pi}_D u = \sum_{i\in I} u_i \chi_{\Omega_i}</math>, where <math>\Omega_i</math> is a dual cell centred on the vertex indexed by <math>i\in I</math>. Using mass lumping allows to get the piecewise constant reconstruction property.
=== Nonconforming
On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others. Then the method enters the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.
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==References==
{{Reflist}}
*'''H. Brezis.''' Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
*'''F. Brezzi, K. Lipnikov, and M. Shashkov.''' Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005.
*'''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.
*'''R. Eymard, C. Guichard, and R. Herbin.''' Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.
*'''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.
*'''K. Lipnikov, G. Manzini, and M. Shaskov.''' Mimetic finite difference method. J. Comput. Phys., 257-Part B:1163–1227, 2014.
*'''G. Strang.''' Variational crimes in the finite element method.'' In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pages 689–710. Academic Press, New York, 1972.
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