2016
DOI: 10.1051/m2an/2015079
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Gradient schemes: Generic tools for the numerical analysis of diffusion equations

Abstract: The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical propertie… Show more

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Cited by 55 publications
(66 citation statements)
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“…as the gradient of the conforming piecewise affine function reconstructed in the triangles from the values at the vertices of the triangles. For proofs that this scheme satisfies the four properties above in the case of homogeneous Dirichlet boundary conditions, see Droniou, Eymard and Herbin [31]. Their adaptation to homogeneous Neumann boundary conditions is once again straightforward (see also [28,Sections 8.3 and 8.4]).…”
Section: Two Examplesmentioning
confidence: 87%
See 1 more Smart Citation
“…as the gradient of the conforming piecewise affine function reconstructed in the triangles from the values at the vertices of the triangles. For proofs that this scheme satisfies the four properties above in the case of homogeneous Dirichlet boundary conditions, see Droniou, Eymard and Herbin [31]. Their adaptation to homogeneous Neumann boundary conditions is once again straightforward (see also [28,Sections 8.3 and 8.4]).…”
Section: Two Examplesmentioning
confidence: 87%
“…For most nonlinear models, convergence proofs using the GDM framework are based on compactness techniques. In contrast to establishing error estimates on the solution -the method favoured by most studies in the literature cited above -such analyses do not require uniqueness or regularity of the solution to the continuous problem, assumptions that are inconsistent with what the physical problem suggests and what the theory provides (see the discussion in Droniou, Eymard and Herbin [31]). The cost of removing these uniqueness/regularity assumptions is the ability to establish rates of convergence with respect to discretisation parameters.…”
Section: Introductionmentioning
confidence: 99%
“…Other gradient discretizations are known for instance in k –conforming finite element methods, 1 –nonconforming finite elements, Multi‐Point Flux Approximations, Discrete Duality Finite Volumes, and Hybrid Mixed Mimetic schemes (which include SUSHI scheme , the Mimetic Finite Difference methods and the Mixed Finite Volume methods ). A full review for these gradient discretizations is provided in . Remark (Some other schemes) . It is shown in that the method of can be identified to the mimetic finite difference method (see ) and the mixed finite volume method .…”
Section: Proof Of Theoremmentioning
confidence: 99%
“…The Gradient Discretization method (GDM) [5,3] provides a common mathematical framework for a number of numerical schemes dedicated to the approximation of elliptic or parabolic problems, linear or nonlinear, coupled or not; these include conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic schemes [4] and some Multi-Point Flux Approximation [1] and Discrete Duality finite volume schemes [2] : we refer to [3, Part III] for more on this (note that in the present proceedings, it is shown that in some way the Discontinuous Galerkin schemes may also enter this framework [6]). Let us recall this framework in the case of the following linear elliptic problem:…”
Section: Introductionmentioning
confidence: 99%