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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 <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
Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to [[#Galerkin methods and conforming finite element methods|conforming Finite Elements]], [[#Mixed finite element|Mixed Finite Elements]],
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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
:<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 GDM is then in this case a nonconforming method for the approximation of (2), which includes the nonconforming finite element method. Note that the reciprocal is not true, in the sense that the GDM framework includes methods such that the function <math>\nabla_D u</math> cannot be computed from the function <math>\Pi_D u</math>.
:<math>\quad (4) \qquad \qquad \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}
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=== Nonconforming finite element ===
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 (these finite elements are used in <ref>'''M. Crouzeix and P.-A. Raviart.''' Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R-3):33–75, 1973.</ref> for the approximation of the Stokes and [[Navier-Stokes equations])). 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.
=== Mixed finite element ===
The [[mixed finite element method]] consists in defining two discrete spaces, one for the approximation of <math>\nabla \overline{u}</math> and another one for <math>\overline{u}</math> <ref>'''P.-A. Raviart and J. M. Thomas.''' A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods ''(Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975)'', pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin, 1977.</ref>.
It suffices to use the discrete relations between these approximations to define a GDM. Using the low degree [[Raviart–Thomas basis functions]] allows to get the piecewise constant reconstruction property.
=== Discontinuous Galerkin method ===
The Discontinuous Galerkin method consists in approximating problems by a piecewise polynomial function, without requirements on the jumps from an element to the other<ref>'''D. A. Di Pietro and A. Ern.''' Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012.</ref>. It is plugged in the GDM by including in the discrete gradient a jump term, acting as the regularization of the gradient in the distribution term.
=== Mimetic finite difference method and nodal mimetic finite difference method ===
This family of methods is introduced by
==See also==
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==References==
{{Reflist}}
▲*'''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.
== External links ==
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