Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

Gerstenmayer, Anita; Jüngel, Ansgar

doi:10.1007/s40314-019-0882-9

Cited by 6 publications

(4 citation statements)

References 36 publications

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“…Indeed, some authors designed finite element/finite difference schemes. Let us mention the following list [8,20,48,50,56,62,63] (again far from being exhaustive) of such contributions. Closer to our study we mention [26], where a model admitting a gradient flow structure for a constraint Wasserstein metric, similar to the one presented in Section 2.1, is discretized thanks to a minimizing JKO scheme [12,59].…”

Section: 3mentioning

confidence: 99%

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A convergent finite volume scheme for dissipation driven models with volume filling constraint

Cancès

Zurek

2022

Numer. Math.

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In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications.

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Section: 3mentioning

confidence: 99%

“…Now our main objective is to adapt the proof of Proposition 9. In this purpose we multiply (50) by log(u k,λ i,K )/d λ i , we sum over K ∈ T and i = 0, . .…”

Section: Proof Of Theoremmentioning

confidence: 99%

A convergent finite volume scheme for dissipation driven models with volume filling constraint

Cancès

Zurek

2022

Numer. Math.

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“…Convergence proofs for finite volume approximations of cross-diffusion systems have been proposed in [3,1,19,15,38,16,21,42,29]. Most of the above contributions rely on the entropy-stability of the schemes, which is exploited thanks to the so-called discrete entropy method [20].…”

Section: It Then Holds That For Allmentioning

confidence: 99%

Finite volumes for the Stefan-Maxwell cross-diffusion system

Cancès¹,

Ehrlacher²,

Monasse³

2020

Preprint

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The aim of this work is to propose a provably convergent finite volume scheme for the so-called Stefan-Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The scheme proposed here relies on a two-point flux approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous models, namely the non-negativity of the solutions, the conservation of mass and the preservation of the volume-filling constraints. In addition, the scheme satisfies a discrete entropy-entropy dissipation relation, very close to the relation which holds at the continuous level. In this article, we present this scheme together with its numerical analysis, and finally illustrate its behaviour with some numerical results.

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“…Until Section 4 we will not make any assumption about the zeros of A. A non-degeneracy assumption will be further assumed in Section 4, but our convergence result could extend to the particular cross-diffusion matrices considered in [38,12,37].…”

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confidence: 99%

A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

Cancès¹,

Gaudeul²

2020

SIAM J. Numer. Anal.

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We study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy dissipation estimate. Numerical results illustrate the behavior of our scheme.

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Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

Cited by 6 publications

References 36 publications

A convergent finite volume scheme for dissipation driven models with volume filling constraint

A convergent finite volume scheme for dissipation driven models with volume filling constraint

Finite volumes for the Stefan-Maxwell cross-diffusion system

A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

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