Convergence of discrete duality finite volume schemes for the cardiac bidomain model

Andreïanov, Boris; Bendahmane, Mostafa; Karlsen, Kenneth H.; Pierre, Charles

doi:10.3934/nhm.2011.6.195

Cited by 15 publications

(33 citation statements)

References 56 publications

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“…We have already pointed out (cf. [6,23]) that discretizations originating form (A),(B) (version of [47]) and (C) coincide, up to details that are not of prime importance. While (D) appears as a quite different method (in particular, the data structure used for implementation is different), a strong connection exists when the primal meshes have quadrangular faces.…”

Section: (D)mentioning

confidence: 88%

“…Numerical results of all these schemes [24,46,47,20,6] show good convergence properties, even on strongly distorted meshes. A direct comparison of these schemes will be held through the 3D anisotropic benchmark [11,21,23] and [42].…”

Section: (D)mentioning

confidence: 99%

“…The reason we focus on the discrete duality features is that they make the finite volume discretization of numerous elliptic operators "structure-preserving"; and they underlie the convergence proofs (see, e.g., [30,9,13,5,20,6] and the sequel [3] of this paper).…”

Section: 2mentioning

confidence: 99%

“…The situation with dual volumes K * can be more intricated: K * may in general consist of a "positive" and a "negative" part, to which no set-theoretical meaning can be given 6 ; but we can give the sense of T ∼K * Vol(T ) to Vol(K * ) . In this case, let us call K * a generalized dual volume.…”

Section: But If (With the Notation Ofmentioning

confidence: 99%

“…From the practical point of view, the method (C) was used in [6] for a further numerical study of the bidomain cardiac model.…”

mentioning

confidence: 99%

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On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

Andreïanov

Bendahmane

Hubert

et al. 2012

IMA Journal of Numerical Analysis

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This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in 3D. Following the so-called DDFV (Disctere Duality Finite Volume) approach developed by F. Hermeline and by K. Domelevo and P. Omnès in 2D, we consider a "double" covering T of a three-dimensional domain by a rather general primal mesh and by a well-chosen "dual" mesh. The associated discrete divergence operator div T is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator ∇ T is defined by local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that −div T , ∇ T are linked by the "discrete duality property", which is an analogue of the integration-by-parts formula. The primal mesh need not be conformal, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel [3] of this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic PDEs.

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Section: (D)mentioning

confidence: 88%

Section: (D)mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: But If (With the Notation Ofmentioning

confidence: 99%

“…From the practical point of view, the method (C) was used in [6] for a further numerical study of the bidomain cardiac model.…”

mentioning

confidence: 99%

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On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

Andreïanov

Bendahmane

Hubert

et al. 2012

IMA Journal of Numerical Analysis

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A discrete duality finite volume discretization of the vorticity‐velocity‐pressure stokes problem on almost arbitrary two‐dimensional grids

Delcourte

2014

Numerical Methods Partial

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We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity‐velocity‐pressure formulation of the two‐dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015

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A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

Bendahmane

Mroue

Saad

2020

Numerical Methods Partial

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The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.

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Convergence of discrete duality finite volume schemes for the cardiac bidomain model

Cited by 15 publications

References 56 publications

On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

A discrete duality finite volume discretization of the vorticity‐velocity‐pressure stokes problem on almost arbitrary two‐dimensional grids

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

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