A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

Hermeline, F.

doi:10.1006/jcph.2000.6466

Cited by 172 publications

(149 citation statements)

References 11 publications

(19 reference statements)

Supporting

Mentioning

143

Contrasting

Unclassified

Order By: Relevance

“…Following Hermeline [57], Domelevo, Omnès [41] and Andreianov, Boyer, Hubert [11], we consider a DDFV mesh which is a triple T = M, M * , S described below.…”

Section: Discrete Duality Finite Volume (Ddfv) Schemesmentioning

confidence: 99%

“…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%

“…Discretization of the aforementioned hyperbolic, porous medium, convection-diffusion, and ellipticparabolic equations by finite volume methods is quite standard by now and often used in engeneering practice. 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.…”

Section: Introductionmentioning

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%

See 4 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

128

275

View full text Add to dashboard Cite

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.

show abstract

“…Following Hermeline [57], Domelevo, Omnès [41] and Andreianov, Boyer, Hubert [11], we consider a DDFV mesh which is a triple T = M, M * , S described below.…”

Section: Discrete Duality Finite Volume (Ddfv) Schemesmentioning

confidence: 99%

“…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 3 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

128

275

View full text Add to dashboard Cite

show abstract

Finite Volume Methods: Foundation and Analysis

Barth

Ohlberger

2004

Encyclopedia of Computational Mechanics

View full text Add to dashboard Cite

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.

show abstract