June  2011, 6(2): 195-240. doi: 10.3934/nhm.2011.6.195

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

1. 

Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

2. 

Université Victor Ségalen - Bordeaux 2, 146 rue Léo Saignat, BP 26, 33076 Bordeaux, France

3. 

Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

4. 

Laboratoire de Mathématiques et Applications, Université de Pau et du Pays de l’Adour, Av. de l’Université, BP 1155, 64013 Pau Cedex,, France

Received  October 2010 Revised  March 2011 Published  May 2011

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.
Citation: Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 183 (1983), 311. doi: 10.1007/BF01176474.

[2]

B. Andreianov, M. Bendahmane, F. Hubert and S. Krell, On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality,, Preprint HAL (2011), (2011).

[3]

B. Andreianov, M. Bendahmane and F. Hubert, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems,, Preprint HAL (2011), (2011).

[4]

B. Andreianov, M. Bendahmane and K. H. Karlsen, A gradient reconstruction formula for finite volume schemes and discrete duality,, In R. Eymard and J.-M. Hérard, (2008), 161.

[5]

B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperbolic Diff. Equ., 7 (2010), 1.

[6]

B. Andreianov, M. Bendahmane and R. Ruiz Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics,, M3AS Math. Models Meth. Appl. Sci., (2011).

[7]

B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes,, Num. Meth. PDE, 23 (2007), 145. doi: 10.1002/num.20170.

[8]

B. Andreianov, M. Gutnic and P. Wittbold, Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: A "continuous" approach,, SIAM J. Num. Anal., 42 (2004), 228. doi: 10.1137/S0036142901400006.

[9]

B. Andreianov, F. Hubert and S. Krell, Benchmark 3D: A version of the DDFV scheme with cell/vertex unknowns on general meshes,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[10]

M. Bendahmane, R. Bürger and R. Ruiz Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities,, Math. Comp. Simul., 80 (2010), 1821. doi: 10.1016/j.matcom.2009.12.010.

[11]

M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue,, Netw. Heterog. Media, 1 (2006), 185.

[12]

M. Bendahmane and K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue,, Appl. Numer. Math., 59 (2009), 2266. doi: 10.1016/j.apnum.2008.12.016.

[13]

S. Börm, L. Grasedyck and W. Hackbusch, An introduction to hierarchical matrices,, Math. Bohemica, 127 (2002), 229.

[14]

S. Börm, L. Grasedyck and W. Hackbusch, Introduction to hierarchical matrices with applications,, Eng. Anal. Bound., 27 (2003), 405. doi: 10.1016/S0955-7997(02)00152-2.

[15]

Y. Bourgault, Y. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electro-physiology,, Nonlin. Anal. Real World Appl., 10 (2009), 458. doi: 10.1016/j.nonrwa.2007.10.007.

[16]

F. Boyer and P. Fabrie, "Eléments d'Analyse pour l'Étude de quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles" (French) [Elements of analysis for the study of some models of incompressible viscous fluid flow],, Math. & Appl. Vol. 52, 52 (2006).

[17]

F. Boyer and F. Hubert, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities,, SIAM J. Num. Anal., 46 (2008), 3032. doi: 10.1137/060666196.

[18]

M. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes,, SIAM J. Num. Anal., 43 (2005), 1872. doi: 10.1137/040613950.

[19]

P. Colli Franzone, L. Guerri and S. Rovida, Wavefront propagation in an activation model of the anisotropic cardiac tissue: Asymptotic analysis and numerical simulations,, J. Math. Biol., 28 (1990), 121. doi: 10.1007/BF00163143.

[20]

P. Colli Franzone, L. Guerri and S. Tentoni, Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field,, Math. Biosci., 101 (1990), 155. doi: 10.1016/0025-5564(90)90020-Y.

[21]

P. Colli Franzone, L. F. Pavarino and B. Taccardi, Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models,, Math. Biosci., 197 (2005), 35. doi: 10.1016/j.mbs.2005.04.003.

[22]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level,, In Evolution equations, 50 (2000), 49.

[23]

Y. Coudière, Th. Gallouët and R. Herbin, Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations,, M2AN Math. Model. Numer. Anal., 35 (2001), 767. doi: 10.1051/m2an:2001135.

[24]

Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equations,, In: A. Handloviovà, (2009), 51.

[25]

Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equation,, HAL preprint (2010), (2010).

[26]

Y. Coudière, F. Hubert and G. Manzini, Benchmark 3D: CeVeFE-DDFV, a discrete duality scheme with cell/vertex/face+edge unknowns,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[27]

Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems,, SIAM J. Numer. Anal., 47 (2010), 4163.

[28]

Y. Coudière and Ch. Pierre, Benchmark 3D: CeVe-DDFV, a discrete duality scheme with cell/vertex unknowns,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[29]

Y. Coudière and Ch. Pierre, Stability and convergence of a finite volume method for two systems of reaction-diffusion in electro-cardiology,, Nonlin. Anal. Real World Appl., 7 (2006), 916. doi: 10.1016/j.nonrwa.2005.02.006.

[30]

Y. Coudière, Ch. Pierre and R. Turpault, A 2D/3D finite volume method used to solve the bidomain equations of electro-cardiology,, Proc. of Algorithmy 2009, (2009).

[31]

Y. Coudière, Ch. Pierre, O. Rousseau and R. Turpault, A 2D/3D discrete duality finite volume scheme. Application to ECG simulation,, Int. J. on Finite Volumes, 6 (2008), 1.

[32]

K. Domelevo, S. Delcourte and P. Omnes, Discrete-duality finite volume method for second order elliptic equations,, in: F. Benkhaldoun, (2005), 447.

[33]

K. Domelevo and P. Omnès., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids,, M2AN Math. Model. Numer. Anal., 39 (2005), 1203. doi: 10.1051/m2an:2005047.

[34]

L. C. Evans, "Partial Differential Equations," vol. 19 of Graduate Studies in Mathematics., American Math. Society, (1998).

[35]

R. Eymard, T. Gallouët and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, VII (2000).

[36]

R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: A scheme using stabilisation and hybrid interfaces,, IMA J. Numer. Anal., 30 (2010), 1009. doi: 10.1093/imanum/drn084.

[37]

R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klöfkorn and G. Manzini, 3D Benchmark on discretization schemes for anisotropic diffusion problems on general grids,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[38]

A. Glitzky and J. A. Griepentrog, Discrete Sobolev-Poincaré inequalities for Voronoï finite volume approximations,, SIAM J. Numer. Anal., 48 (2010), 372. doi: 10.1137/09076502X.

[39]

D. Harrild and C. S. Henriquez, A finite volume model of cardiac propagation,, Ann. Biomed. Engrg., 25 (1997), 315. doi: 10.1007/BF02648046.

[40]

R. Herbin and F. Hubert, Benchmark on discretisation schemes for anisotropic diffusion problems on general grids,, In R. Eymard and J.-M. Hérard, (2008), 659.

[41]

C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the biodomain models,, Crit. Rev. Biomed. Engr., 21 (1993), 1.

[42]

F. Hermeline, Une méthode de volumes finis pour les équations elliptiques du second ordre (French) [A finite-volume method for second-order elliptic equations],, C. R. Math. Acad. Sci. Paris Sér. I, 326 (1198), 1433.

[43]

F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes,, J. Comput. Phys., 160 (2000), 481. doi: 10.1006/jcph.2000.6466.

[44]

F. Hermeline, A finite volume method for solving Maxwell equations in inhomogeneous media on arbitrary meshes,, C. R. Math. Acad. Sci. Paris Sér. I, 339 (2004), 893.

[45]

F. Hermeline, Approximation of 2D and 3D diffusion operators with discontinuous full-tensor coefficients on arbitrary meshes,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2497. doi: 10.1016/j.cma.2007.01.005.

[46]

F. Hermeline, A finite volume method for approximating 3D diffusion operators on general meshes,, J. Comput. Phys., 228 (2009), 5763. doi: 10.1016/j.jcp.2009.05.002.

[47]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.

[48]

J. Keener and J. Sneyd, "Mathematical Physiology," Vol. 8 of, Interdisciplinary Applied Mathematics, (1998).

[49]

S. Krell, Stabilized DDFV schemes for Stokes problem with variable viscosity on general 2D meshes,, Num. Meth. PDEs, (2010).

[50]

S. Krell and G. Manzini, The Discrete Duality Finite Volume method for the Stokes equations on 3D polyhedral meshes,, HAL preprint (2010), (2010).

[51]

S. N. Kruzhkov, Results on the nature of the continuity of solutions of parabolic equations and some of their applications,, Mat. Zametki, 6 (1969), 97.

[52]

P. Le Guyader, F. Trelles and P. Savard, Extracellular measurement of anisotropic bidomain myocardial conductivities. I. Theoretical analysis,, Annals Biomed. Eng., 29 (2001), 862. doi: 10.1114/1.1408923.

[53]

G. T. Lines, P. Grottum, A. J. Pullan, J. Sundes and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology,, Comput. Visual. Sci., 5 (2002), 215.

[54]

G. Lines, M. L. Buist, P. Grøttum, A. J. Pullan, J. Sundnes and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology,, Comput. Visual. Sci., 5 (2003), 215.

[55]

J.-L. Lions and E. Magenes, "Problèmes aux Limites non Homogènes et Applications," Vol. 1, (French) [Nonhomogeneous boundary value problems and their applications. Vol. 1],, Dunod, (1968).

[56]

C.-H. Luo and Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction,, Circ. Res., 68 (1991), 1501.

[57]

D. Noble, A modification of the Hodgkin-Huxley equation applicable to Purkinje fibre action and pacemaker potentials,, J. Physiol., 160 (1962), 317.

[58]

F. Otto, $L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations,, J. Diff. Equ., 131 (1996), 20. doi: 10.1006/jdeq.1996.0155.

[59]

Ch. Pierre, "Modélisation et Simulation de l'Activité Électrique du Coeur dans le Thorax, Analyse Numérique et Méthodes de Volumes Finis" (French) [Modelling and Simulation of the Heart Electrical Activity in the Thorax, Numerical Analysis and Finite Volume Methods], Ph.D. Thesis, (2005).

[60]

Ch. Pierre, Preconditioning the coupled heart and torso bidomain model with an almost linear complexity,, HAL Preprint (2010), (2010).

[61]

S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electro-cardiology,, Numer. Meth. PDE, 18 (2002), 218. doi: 10.1002/num.1000.

[62]

J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K.-A. Mardal and A. Tveito, "Computing the Electrical Activity in the Human Heart,", Springer, (2005).

[63]

J. Sundnes, G. T. Lines and A. Tveito, An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso,, Math. Biosci., 194 (2005), 233. doi: 10.1016/j.mbs.2005.01.001.

[64]

L. Tung, "A Bidomain Model for Describing Ischemic Myocardial D-D Properties,", Ph.D. thesis, (1978).

[65]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field,, Math. Methods Appl. Sci., 29 (2006), 1631. doi: 10.1002/mma.740.

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 183 (1983), 311. doi: 10.1007/BF01176474.

[2]

B. Andreianov, M. Bendahmane, F. Hubert and S. Krell, On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality,, Preprint HAL (2011), (2011).

[3]

B. Andreianov, M. Bendahmane and F. Hubert, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems,, Preprint HAL (2011), (2011).

[4]

B. Andreianov, M. Bendahmane and K. H. Karlsen, A gradient reconstruction formula for finite volume schemes and discrete duality,, In R. Eymard and J.-M. Hérard, (2008), 161.

[5]

B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperbolic Diff. Equ., 7 (2010), 1.

[6]

B. Andreianov, M. Bendahmane and R. Ruiz Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics,, M3AS Math. Models Meth. Appl. Sci., (2011).

[7]

B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes,, Num. Meth. PDE, 23 (2007), 145. doi: 10.1002/num.20170.

[8]

B. Andreianov, M. Gutnic and P. Wittbold, Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: A "continuous" approach,, SIAM J. Num. Anal., 42 (2004), 228. doi: 10.1137/S0036142901400006.

[9]

B. Andreianov, F. Hubert and S. Krell, Benchmark 3D: A version of the DDFV scheme with cell/vertex unknowns on general meshes,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[10]

M. Bendahmane, R. Bürger and R. Ruiz Baier, A finite volume scheme for cardiac propagation in media with isotropic conductivities,, Math. Comp. Simul., 80 (2010), 1821. doi: 10.1016/j.matcom.2009.12.010.

[11]

M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue,, Netw. Heterog. Media, 1 (2006), 185.

[12]

M. Bendahmane and K. H. Karlsen, Convergence of a finite volume scheme for the bidomain model of cardiac tissue,, Appl. Numer. Math., 59 (2009), 2266. doi: 10.1016/j.apnum.2008.12.016.

[13]

S. Börm, L. Grasedyck and W. Hackbusch, An introduction to hierarchical matrices,, Math. Bohemica, 127 (2002), 229.

[14]

S. Börm, L. Grasedyck and W. Hackbusch, Introduction to hierarchical matrices with applications,, Eng. Anal. Bound., 27 (2003), 405. doi: 10.1016/S0955-7997(02)00152-2.

[15]

Y. Bourgault, Y. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electro-physiology,, Nonlin. Anal. Real World Appl., 10 (2009), 458. doi: 10.1016/j.nonrwa.2007.10.007.

[16]

F. Boyer and P. Fabrie, "Eléments d'Analyse pour l'Étude de quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles" (French) [Elements of analysis for the study of some models of incompressible viscous fluid flow],, Math. & Appl. Vol. 52, 52 (2006).

[17]

F. Boyer and F. Hubert, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities,, SIAM J. Num. Anal., 46 (2008), 3032. doi: 10.1137/060666196.

[18]

M. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes,, SIAM J. Num. Anal., 43 (2005), 1872. doi: 10.1137/040613950.

[19]

P. Colli Franzone, L. Guerri and S. Rovida, Wavefront propagation in an activation model of the anisotropic cardiac tissue: Asymptotic analysis and numerical simulations,, J. Math. Biol., 28 (1990), 121. doi: 10.1007/BF00163143.

[20]

P. Colli Franzone, L. Guerri and S. Tentoni, Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field,, Math. Biosci., 101 (1990), 155. doi: 10.1016/0025-5564(90)90020-Y.

[21]

P. Colli Franzone, L. F. Pavarino and B. Taccardi, Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models,, Math. Biosci., 197 (2005), 35. doi: 10.1016/j.mbs.2005.04.003.

[22]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level,, In Evolution equations, 50 (2000), 49.

[23]

Y. Coudière, Th. Gallouët and R. Herbin, Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations,, M2AN Math. Model. Numer. Anal., 35 (2001), 767. doi: 10.1051/m2an:2001135.

[24]

Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equations,, In: A. Handloviovà, (2009), 51.

[25]

Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equation,, HAL preprint (2010), (2010).

[26]

Y. Coudière, F. Hubert and G. Manzini, Benchmark 3D: CeVeFE-DDFV, a discrete duality scheme with cell/vertex/face+edge unknowns,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[27]

Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems,, SIAM J. Numer. Anal., 47 (2010), 4163.

[28]

Y. Coudière and Ch. Pierre, Benchmark 3D: CeVe-DDFV, a discrete duality scheme with cell/vertex unknowns,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[29]

Y. Coudière and Ch. Pierre, Stability and convergence of a finite volume method for two systems of reaction-diffusion in electro-cardiology,, Nonlin. Anal. Real World Appl., 7 (2006), 916. doi: 10.1016/j.nonrwa.2005.02.006.

[30]

Y. Coudière, Ch. Pierre and R. Turpault, A 2D/3D finite volume method used to solve the bidomain equations of electro-cardiology,, Proc. of Algorithmy 2009, (2009).

[31]

Y. Coudière, Ch. Pierre, O. Rousseau and R. Turpault, A 2D/3D discrete duality finite volume scheme. Application to ECG simulation,, Int. J. on Finite Volumes, 6 (2008), 1.

[32]

K. Domelevo, S. Delcourte and P. Omnes, Discrete-duality finite volume method for second order elliptic equations,, in: F. Benkhaldoun, (2005), 447.

[33]

K. Domelevo and P. Omnès., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids,, M2AN Math. Model. Numer. Anal., 39 (2005), 1203. doi: 10.1051/m2an:2005047.

[34]

L. C. Evans, "Partial Differential Equations," vol. 19 of Graduate Studies in Mathematics., American Math. Society, (1998).

[35]

R. Eymard, T. Gallouët and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, VII (2000).

[36]

R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: A scheme using stabilisation and hybrid interfaces,, IMA J. Numer. Anal., 30 (2010), 1009. doi: 10.1093/imanum/drn084.

[37]

R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klöfkorn and G. Manzini, 3D Benchmark on discretization schemes for anisotropic diffusion problems on general grids,, In Proc. of Finite Volumes for Complex Applications VI in Prague, (2011).

[38]

A. Glitzky and J. A. Griepentrog, Discrete Sobolev-Poincaré inequalities for Voronoï finite volume approximations,, SIAM J. Numer. Anal., 48 (2010), 372. doi: 10.1137/09076502X.

[39]

D. Harrild and C. S. Henriquez, A finite volume model of cardiac propagation,, Ann. Biomed. Engrg., 25 (1997), 315. doi: 10.1007/BF02648046.

[40]

R. Herbin and F. Hubert, Benchmark on discretisation schemes for anisotropic diffusion problems on general grids,, In R. Eymard and J.-M. Hérard, (2008), 659.

[41]

C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the biodomain models,, Crit. Rev. Biomed. Engr., 21 (1993), 1.

[42]

F. Hermeline, Une méthode de volumes finis pour les équations elliptiques du second ordre (French) [A finite-volume method for second-order elliptic equations],, C. R. Math. Acad. Sci. Paris Sér. I, 326 (1198), 1433.

[43]

F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes,, J. Comput. Phys., 160 (2000), 481. doi: 10.1006/jcph.2000.6466.

[44]

F. Hermeline, A finite volume method for solving Maxwell equations in inhomogeneous media on arbitrary meshes,, C. R. Math. Acad. Sci. Paris Sér. I, 339 (2004), 893.

[45]

F. Hermeline, Approximation of 2D and 3D diffusion operators with discontinuous full-tensor coefficients on arbitrary meshes,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2497. doi: 10.1016/j.cma.2007.01.005.

[46]

F. Hermeline, A finite volume method for approximating 3D diffusion operators on general meshes,, J. Comput. Phys., 228 (2009), 5763. doi: 10.1016/j.jcp.2009.05.002.

[47]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500.

[48]

J. Keener and J. Sneyd, "Mathematical Physiology," Vol. 8 of, Interdisciplinary Applied Mathematics, (1998).

[49]

S. Krell, Stabilized DDFV schemes for Stokes problem with variable viscosity on general 2D meshes,, Num. Meth. PDEs, (2010).

[50]

S. Krell and G. Manzini, The Discrete Duality Finite Volume method for the Stokes equations on 3D polyhedral meshes,, HAL preprint (2010), (2010).

[51]

S. N. Kruzhkov, Results on the nature of the continuity of solutions of parabolic equations and some of their applications,, Mat. Zametki, 6 (1969), 97.

[52]

P. Le Guyader, F. Trelles and P. Savard, Extracellular measurement of anisotropic bidomain myocardial conductivities. I. Theoretical analysis,, Annals Biomed. Eng., 29 (2001), 862. doi: 10.1114/1.1408923.

[53]

G. T. Lines, P. Grottum, A. J. Pullan, J. Sundes and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology,, Comput. Visual. Sci., 5 (2002), 215.

[54]

G. Lines, M. L. Buist, P. Grøttum, A. J. Pullan, J. Sundnes and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology,, Comput. Visual. Sci., 5 (2003), 215.

[55]

J.-L. Lions and E. Magenes, "Problèmes aux Limites non Homogènes et Applications," Vol. 1, (French) [Nonhomogeneous boundary value problems and their applications. Vol. 1],, Dunod, (1968).

[56]

C.-H. Luo and Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction,, Circ. Res., 68 (1991), 1501.

[57]

D. Noble, A modification of the Hodgkin-Huxley equation applicable to Purkinje fibre action and pacemaker potentials,, J. Physiol., 160 (1962), 317.

[58]

F. Otto, $L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations,, J. Diff. Equ., 131 (1996), 20. doi: 10.1006/jdeq.1996.0155.

[59]

Ch. Pierre, "Modélisation et Simulation de l'Activité Électrique du Coeur dans le Thorax, Analyse Numérique et Méthodes de Volumes Finis" (French) [Modelling and Simulation of the Heart Electrical Activity in the Thorax, Numerical Analysis and Finite Volume Methods], Ph.D. Thesis, (2005).

[60]

Ch. Pierre, Preconditioning the coupled heart and torso bidomain model with an almost linear complexity,, HAL Preprint (2010), (2010).

[61]

S. Sanfelici, Convergence of the Galerkin approximation of a degenerate evolution problem in electro-cardiology,, Numer. Meth. PDE, 18 (2002), 218. doi: 10.1002/num.1000.

[62]

J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K.-A. Mardal and A. Tveito, "Computing the Electrical Activity in the Human Heart,", Springer, (2005).

[63]

J. Sundnes, G. T. Lines and A. Tveito, An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso,, Math. Biosci., 194 (2005), 233. doi: 10.1016/j.mbs.2005.01.001.

[64]

L. Tung, "A Bidomain Model for Describing Ischemic Myocardial D-D Properties,", Ph.D. thesis, (1978).

[65]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field,, Math. Methods Appl. Sci., 29 (2006), 1631. doi: 10.1002/mma.740.

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