Gradient discretisation method: Difference between revisions

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{{Short description|Method for numerical differential equations}}

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[[Image:Plaplacien4.svg|thumb|right|400px|Exact solution <br/> <math>\overline{u}(x) = \frac 3 4 \left({0.5}^{4/3}- |x - 0.5|^{4/3}\right)</math> <br/>

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(black line) and approximate one (blue line) computed with the first degree discontinuous Galerkin method plugged into the GDM (uniform mesh with 6 elements).]]

{{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 [[Raphaèle Herbin|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 [[Raphaèle Herbin|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>

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{{NumBlk|:|<math>\forall v \in X_{D,0},\qquad \int_{\Omega} \nabla_D u(x)\cdot\nabla_D v(x) dx = \int_{\Omega} f(x)\Pi_D v(x) dx. </math>|{{EquationRef|3}}}}

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 reciprocalconverse 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>.

The following error estimate, inspired by G. Strang's second lemma,<ref>'''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.</ref> holds