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

Mostafa Bendahmane,Boris Andreianov,,Laboratoire De Mathématiques Cnrs Umr 6623, Université De Franche-Comté, 16 Route De Gray, 25030 Besançon Cedex ,,Laboratoire De Mathématiques Et Applications, Université De Pau Et Du Pays De L’adour, Av. De L’université, Bp 1155, 64013 Pau Cedex, ,Charles Pierre,,Université Victor Ségalen - Bordeaux 2, 146 Rue Léo Saignat, Bp 26, 33076 Bordeaux ,Kenneth H. Karlsen,,Centre Of Mathematics For Applications, University Of Oslo, P.o. Box 1053, Blindern, N–0316 Oslo

doi:10.3934/nhm.2011.6.195

Abstract

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Highlights

We consider the heart of a living organism that occupies a fixed domain Ω, which is assumed to be a bounded open subset of R3 with Lipschitz boundary ∂Ω
Our study can be considered as a theoretical and numerical validation of the discrete duality finite volume (DDFV) discretisation strategy for the bidomain model. For both a fully timeimplicit scheme and a linearised time-implicit scheme, we prove convergence of different DDFV discretisations to the unique solution of the bidomain model (1)
We prove the convergence of each of these schemes, using only general properties of DDFV approximations; but the main focus is on scheme (B), which construction is detailed

Summary

Introduction

We consider the heart of a living organism that occupies a fixed domain Ω, which is assumed to be a bounded open subset of R3 with Lipschitz boundary ∂Ω. 2our notation follows [7]; a slightly different viewpoint was used in [4, 5, 2, 3], where the homogeneous Dirichlet boundary data were included into the definition of the space RT 0 of discrete functions defined on the control volumes adjacent to ΓD ≡ ∂Ω. In this paper a standard finite volume divergence operator is defined and the definition of a gradient operator is replaced by a dual property (of discrete Green Gauss formula type) together with the definition of a consistent scalar product. A major difference is that, the discrete divergence need not be computed in practice, it has a simple and natural definition (it is a flux balance as usual for finite volume schemes), whereas in the case of Mimetic schemes only a dual and abstract definition of the gradient is provided.

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