Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes

Hermeline, F.

doi:10.1016/s0045-7825(02)00644-8

Cited by 82 publications

(88 citation statements)

References 10 publications

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“…On the other hand, if this term is dropped, non-conformal double meshes can be considered for d = 2 (see [57,41,11]) and d = 3 (see [72,39,58,8,7,38]) within the variational framework.…”

Section: Discrete Duality Finite Volume (Ddfv) Schemesmentioning

confidence: 99%

“…We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]).…”

Section: Introductionmentioning

confidence: 99%

“…We adapt the approximations used by Eymard, Gallouët, Herbin [48] (see also [31,79,49,67]) for the nonlinear convection term, and those used by Hermeline [57,58], Domelevo, Omnès [41] and Andreianov, Boyer, Hubert [11] for the doubly nonlinear diffusion term. In 3D, we propose new DDFV schemes that possess convenient discrete duality properties.…”

Section: Introductionmentioning

confidence: 99%

“…Our 3D scheme is a very particular case of the schemes introduced and studied numerically by Hermeline in [58]. In passing, we mention that different kinds of 3D discrete duality schemes were constructed in [72,39] and in [38].…”

Section: Introductionmentioning

confidence: 99%

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Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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“…On the other hand, if this term is dropped, non-conformal double meshes can be considered for d = 2 (see [57,41,11]) and d = 3 (see [72,39,58,8,7,38]) within the variational framework.…”

Section: Discrete Duality Finite Volume (Ddfv) Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

View full text Add to dashboard Cite

show abstract

“…The principle of the second family, the so-called "vertex-centered" schemes, is to associate discrete unknowns with the vertices of the primal mesh, and then integrate the Laplace equation on the cells of a dual mesh, centered on the vertices [4,5,10,11,24,35]. More recently, a third family of schemes has emerged, which combines the previous two approaches, since these schemes associate unknowns with both the cells and the vertices of the mesh, and integrate the Laplace equation on both the cells of the primal and dual meshes [3,13,16,18,19,26,27,33]. The originality of these schemes is that they work well on all kind of meshes, including very distorted, degenerating, or highly nonconforming meshes (see the numerical tests in [19]).…”

Section: Introductionmentioning

confidence: 99%

On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Omnes

2010

ESAIM: M2AN

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Abstract. Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P 1 function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L 2 norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1 (Ω).Mathematics Subject Classification. 65N15, 65N30, 35J05.

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Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes

Cited by 82 publications

References 10 publications

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Finite Volume Methods: Foundation and Analysis

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