A cell-centered diffusion scheme on two-dimensional unstructured meshes

Breil, J.; Maire, Pierre‐Henri

doi:10.1016/j.jcp.2006.10.025

Cited by 103 publications

(99 citation statements)

References 23 publications

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“…Considering a patch of (sub)elements, supposing a piecewise affine, possibly nonconforming, potential approximation, imposing the normal flux continuity, and solving a local linear system on the patch is also the principle of the MPFA method [1,3,19], cf. also [28,41,14,56,42]. In the MPFA method, the fluxes on patches of (sub)elements are locally recovered from the potentials strongly by u h := −S∇p h .…”

Section: Definition Of the Local Problems By A Geometrical Interpretamentioning

confidence: 99%

“…The last goal of the present paper is to recall known and show new relations between MFE methods and other discretization schemes, namely the two-point finite volume (FV) [29], mimetic finite difference (MFD) [17,13], hybrid finite volume [31], mixed finite volume [26], multi-point flux-approximation (MPFA) [1,3,19], and related methods [28,41,14,56,42]. We in particular prove in Section 7 that the RT0 MFE method on arbitrary polygonal meshes is a particular example of the MFD method.…”

Section: Introductionmentioning

confidence: 99%

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Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

Vohralı́k

Wohlmuth

2013

Math. Models Methods Appl. Sci.

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In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart-Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart-Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

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Section: Definition Of the Local Problems By A Geometrical Interpretamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

Vohralı́k

Wohlmuth

2013

Math. Models Methods Appl. Sci.

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“…In order to validate the theory, 2D simulations have been carried out with the code CHIC [9]. These simulations take a 20 m thick SiO 2 layer irradiated by a laser pulse with a maximum intensity of about 04008-p. 4 …”

Section: Numerical Self-consistent Model For Stability Analysismentioning

confidence: 99%

Modeling hydrodynamic instabilities of double ablation fronts in inertial confinement fusion

Yañez

Sanz

Olazabal-Loumé

et al. 2013

EPJ Web of Conferences

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Abstract.A linear Rayleigh-Taylor instability theory of double ablation (DA) fronts is developed for directdrive inertial confinement fusion. Two approaches are discussed: an analytical discontinuity model for the radiation dominated regime of very steep DA front structure, and a numerical self-consistent model that covers more general hydrodynamic profiles behaviours. Dispersion relation results are compared to 2D simulations.

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“…We observe that some authors mentioned linearity preserving criterion in their works [19,21,20,5,6]. To our knowledge, the authors usually claimed that certain scheme satisfies or does not satisfy the linearity preserving criterion without intensive investigation.…”

mentioning

confidence: 90%

A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

Gao

2010

Numerical Methods in Fluids

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Abstract. In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell-centered unknowns, which reduces the scheme to a completely cell-centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity preserving cell-centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.

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A cell-centered diffusion scheme on two-dimensional unstructured meshes

Cited by 103 publications

References 23 publications

Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

Modeling hydrodynamic instabilities of double ablation fronts in inertial confinement fusion

A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

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