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 discretisationnumerical schemes for diffusion problems of differentvarious 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. Owing to theseThese core properties, itenable iscomplete possibleproofs to prove theof convergence of athe GDM for standard elliptic and parabolic problems, linear or non-linear. AsFor alinear consequenceproblems, anystationary schemeor enteringtransient, theerror GDMestimates frameworkcan isbe thenestablished knownbased on three indicators specific to convergethe onGDM these(the problems;quantities this<math>C_{D}</math>, occurs<math>S_{D}</math> inand the<math>W_{D}</math>, case[[#The example of thea conforminglinear Finitediffusion Elementsproblem|see below]]). For non-linear problems, the Raviart—Thomasproofs Mixedare Finitebased Elements,on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the <math>P^1</math>model data. [[#Some non-conforminglinear Finiteproblems Elements,with orcomplete inconvergence proofs of the caseGDM|Non-linear models]] for which such convergence proof of morethe recentGDM schemes,have suchbeen ascarried out comprise: the HybridStefan Mixedmodel Mimeticof ora Nodalmelting Mimeticmaterial, two-phase flows in porous methodsmedia, somethe DiscreteRichards Dualityequation Finiteof underground Volumewater schemesflow, andthe somefully Multinon-Point Fluxlinear ApproximationLeray—Lions schemesequations.

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>,. andThe theusual solutionsense <math>\overline{u}\inof H^1_0(\Omega)</math>weak issolution suchto thatthis model is:

:<math>\quad (2) \qquad\mbox{Find }\qquadoverline{u}\in H^1_0(\forallOmega)\mbox{ such that, for all } \overline{v} \in H^1_0(\Omega),\qquadquad \int_{\Omega} \nabla \overline{u}(x)\cdot\nabla \overline{v}(x) dx = \int_{\Omega} f(x)\overline{v}(x) dx. </math>

AIn a nutshell, the GDM for such a model consists in selecting a finite-dimensional space and two reconstruction operators (one for the functions, one for the gradients) and to substitute these discrete elements in lieu of the continuous elements in (2). More precisely, the GDM starts by defining a Gradient Discretization (GD), iswhich defined byis a triplet <math>D = (X_{D,0},\Pi_D,\nabla_D)</math>, where:

* the set of discrete unknowns <math>X_{D,0}</math> is a finite dimensional real vector space,

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The operator <math>\Pi_D</math> is a piecewise constant reconstruction if there exists a basis <math>(e_i)_{i\in B}</math> of <math>X_{D,0}</math> and a family of disjoint subsets <math>(\Omega_i)_{i\in B}</math> of <math>\Omega</math> such that <math>\Pi_D u = \sum_{i\in B}u_i\chi_{\Omega_i}</math> for all <math>u=\sum_{i\in B} u_i e_i\in X_{D,0}</math>, where <math>\chi_{\Omega_i}</math> is the characteristic function of <math>\Omega_i</math>.

==ReviewSome of somenon-linear problems whichwith maycomplete beconvergence approximatedproofs byof athe GDM==

We review some problems for which the GDM can be proved to converge when the above core properties are satisfied.