# American Institute of Mathematical Sciences

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. Google Scholar [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). Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] S. Börm, L. Grasedyck and W. Hackbusch, An introduction to hierarchical matrices,, Math. Bohemica, 127 (2002), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equations,, In: A. Handloviovà, (2009), 51. Google Scholar [25] Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equation,, HAL preprint (2010), (2010). Google Scholar [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). Google Scholar [27] Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems,, SIAM J. Numer. Anal., 47 (2010), 4163. Google Scholar [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). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [32] K. Domelevo, S. Delcourte and P. Omnes, Discrete-duality finite volume method for second order elliptic equations,, in: F. Benkhaldoun, (2005), 447. Google Scholar [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. Google Scholar [34] L. C. Evans, "Partial Differential Equations," vol. 19 of Graduate Studies in Mathematics., American Math. Society, (1998). Google Scholar [35] R. Eymard, T. Gallouët and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, VII (2000). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [39] D. Harrild and C. S. Henriquez, A finite volume model of cardiac propagation,, Ann. Biomed. Engrg., 25 (1997), 315. doi: 10.1007/BF02648046. Google Scholar [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. Google Scholar [41] C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the biodomain models,, Crit. Rev. Biomed. Engr., 21 (1993), 1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] J. Keener and J. Sneyd, "Mathematical Physiology," Vol. 8 of, Interdisciplinary Applied Mathematics, (1998). Google Scholar [49] S. Krell, Stabilized DDFV schemes for Stokes problem with variable viscosity on general 2D meshes,, Num. Meth. PDEs, (2010). Google Scholar [50] S. Krell and G. Manzini, The Discrete Duality Finite Volume method for the Stokes equations on 3D polyhedral meshes,, HAL preprint (2010), (2010). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [57] D. Noble, A modification of the Hodgkin-Huxley equation applicable to Purkinje fibre action and pacemaker potentials,, J. Physiol., 160 (1962), 317. Google Scholar [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. Google Scholar [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). Google Scholar [60] Ch. Pierre, Preconditioning the coupled heart and torso bidomain model with an almost linear complexity,, HAL Preprint (2010), (2010). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [64] L. Tung, "A Bidomain Model for Describing Ischemic Myocardial D-D Properties,", Ph.D. thesis, (1978). Google Scholar [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. Google Scholar

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. Google Scholar [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). Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] S. Börm, L. Grasedyck and W. Hackbusch, An introduction to hierarchical matrices,, Math. Bohemica, 127 (2002), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equations,, In: A. Handloviovà, (2009), 51. Google Scholar [25] Y. Coudière and F. Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equation,, HAL preprint (2010), (2010). Google Scholar [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). Google Scholar [27] Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems,, SIAM J. Numer. Anal., 47 (2010), 4163. Google Scholar [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). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [32] K. Domelevo, S. Delcourte and P. Omnes, Discrete-duality finite volume method for second order elliptic equations,, in: F. Benkhaldoun, (2005), 447. Google Scholar [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. Google Scholar [34] L. C. Evans, "Partial Differential Equations," vol. 19 of Graduate Studies in Mathematics., American Math. Society, (1998). Google Scholar [35] R. Eymard, T. Gallouët and R. Herbin, "Finite Volume Methods,", Handbook of Numerical Analysis, VII (2000). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [39] D. Harrild and C. S. Henriquez, A finite volume model of cardiac propagation,, Ann. Biomed. Engrg., 25 (1997), 315. doi: 10.1007/BF02648046. Google Scholar [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. Google Scholar [41] C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the biodomain models,, Crit. Rev. Biomed. Engr., 21 (1993), 1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] J. Keener and J. Sneyd, "Mathematical Physiology," Vol. 8 of, Interdisciplinary Applied Mathematics, (1998). Google Scholar [49] S. Krell, Stabilized DDFV schemes for Stokes problem with variable viscosity on general 2D meshes,, Num. Meth. PDEs, (2010). Google Scholar [50] S. Krell and G. Manzini, The Discrete Duality Finite Volume method for the Stokes equations on 3D polyhedral meshes,, HAL preprint (2010), (2010). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [57] D. Noble, A modification of the Hodgkin-Huxley equation applicable to Purkinje fibre action and pacemaker potentials,, J. Physiol., 160 (1962), 317. Google Scholar [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. Google Scholar [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). Google Scholar [60] Ch. Pierre, Preconditioning the coupled heart and torso bidomain model with an almost linear complexity,, HAL Preprint (2010), (2010). Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [64] L. Tung, "A Bidomain Model for Describing Ischemic Myocardial D-D Properties,", Ph.D. thesis, (1978). Google Scholar [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. Google Scholar
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