Free Access
Issue
ESAIM: M2AN
Volume 47, Number 2, March-April 2013
Page(s) 583 - 608
DOI https://doi.org/10.1051/m2an/2012040
Published online 18 January 2013
  1. LifeV software. http://www.LifeV.org. [Google Scholar]
  2. Trilinos software. http://trilinos.sandia.gov. [Google Scholar]
  3. A. Alonso-Rodriguez and L. Gerardo-Giorda, New non-overlapping domain decomposition methods for the time-harmonic Maxwell system. SIAM J. Sci. Comput. 28 (2006) 102–122. [CrossRef] [Google Scholar]
  4. P. Bochev and R. Lehouc, On the finite element solution of the pure Neumann problem. SIAM Rev. 47 (2005) 50–66. [CrossRef] [MathSciNet] [Google Scholar]
  5. T.F. Chan and T.P. Mathew, Domain decomposition algorithms, in Acta Numerica 1994. Cambridge University Press (1994) 61–143. [Google Scholar]
  6. P. Charton, F. Nataf and F. Rogier, Méthode de décomposition de domaine pour l’équation d’advection-diffusion. C. R. Acad. Sci. 313 (1991) 623–626. [Google Scholar]
  7. P. Chevalier and F. Nataf, Symmetrized method with optimized second-order conditions for the Helmholtz equation, in Domain decomposition methods (Boulder, CO, 1997). Amer. Math. Soc. 10 (1998) 400–407. [Google Scholar]
  8. R.H. Clayton, O.M. Bernus, E.M. Cherry, H. Dierckx, F.H. Fenton, L. Mirabella, A.V. Panfilov, F.B. Sachse, G. Seemann and H. Zhang, Models of cardiac tissue electrophysiology : Progress, challenges and open questions. Progr. Bioph. Molec. Biol. 104 (2011) 22–48. [CrossRef] [Google Scholar]
  9. R.H. Clayton and A.V. Panfilov, A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Progr. Bioph. Molec. Biol. 96 (2008) 19–43. [CrossRef] [PubMed] [Google Scholar]
  10. P. Colli Franzone and L.F. Pavarino, A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14 (2004) 883–911. [CrossRef] [Google Scholar]
  11. P. Colli Franzone, L. Pavarino and G. Savaré, Computational electrocardiology : mathematical and numerical modeling, in Complex Systems in Biomedicine – A. Quarteroni, edited by L. Formaggia and A. Veneziani. Springer, Milan (2006). [Google Scholar]
  12. P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, edited by A. Lorenzi and B. Ruf. Birkhauser (2002) 49–78. [Google Scholar]
  13. Q. Deng, An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput. 18 (1997) 1517–1525. [CrossRef] [Google Scholar]
  14. V. Dolean and F. Nataf, An Optimized Schwarz Algorithm for the compressible Euler equations, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 173–180. [Google Scholar]
  15. V. Dolean, M.J. Gander and L. Gerardo-Giorda, Optimized Schwarz Methods for Maxwell’s equations. SIAM J. Sci. Comput. 31 (2009) 2193–2213. [CrossRef] [MathSciNet] [Google Scholar]
  16. O. Dubois, Optimized Schwarz Methods with Robin conditions for the Advection-Diffusion Equation, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 181–188. [Google Scholar]
  17. B. Engquist and H.-K. Zhao, Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341–365. [CrossRef] [MathSciNet] [Google Scholar]
  18. E. Faccioli, F. Maggio, A. Quarteroni and A. Tagliani, Spectral domain decomposition methods for the solution of acoustic and elastic wave propagation. Geophys. 61 (1996) 1160–1174. [CrossRef] [Google Scholar]
  19. E. Faccioli, F. Maggio, A. Quarteroni and A. Tagliani, 2d and 3d elastic wave propagation by pseudo-spectral domain decomposition method. J. Seismology 1 (1997) 237–251. [CrossRef] [Google Scholar]
  20. M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. [CrossRef] [MathSciNet] [Google Scholar]
  21. M.J. Gander and L. Halpern, Méthodes de relaxation d’ondes pour l’équation de la chaleur en dimension 1. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 519–524. [CrossRef] [Google Scholar]
  22. M.J. Gander, L. Halpern and F. Magoulès, An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Meth. Fluids 55 (2007) 163–175. [CrossRef] [MathSciNet] [Google Scholar]
  23. M.J. Gander, L. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643–1681. [CrossRef] [MathSciNet] [Google Scholar]
  24. M.J. Gander, F. Magoulès and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38–60. [CrossRef] [MathSciNet] [Google Scholar]
  25. L. Gerardo-Giorda, L. Mirabella, M. Perego and A. Veneziani, A model adaptive strategy for computational electrocardiology. Domain Decomposition Methods in Science and Engineering XXI (Proceedings of the DD21 Conference). Springer-Verlag. To appear (2013). [Google Scholar]
  26. L. Gerardo-Giorda, L. Mirabella, F. Nobile, M. Perego and A. Veneziani, A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 3625–3639. [CrossRef] [Google Scholar]
  27. L. Gerardo-Giorda, F. Nobile and C. Vergara, Analysis and optimization of Robin–Robin partitioned procedures in Fluid-Structure Interaction problems. SIAM J. Numer. Anal. 48 (2010) 2091–2116. [CrossRef] [MathSciNet] [Google Scholar]
  28. L. Gerardo-Giorda, M. Perego and A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. ESAIM : M2AN 45 (2011) 309–334. [CrossRef] [EDP Sciences] [Google Scholar]
  29. T. Hagstrom, R.P. Tewarson and A. Jazcilevich, Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett. 1 (1988) 299–302. [CrossRef] [Google Scholar]
  30. C.S. Henriquez, Simulating the electrical behavior of cardiac tissue using the Bidomain model. Crit. Rev. Biomed. Eng. 21 (1993) 1–77. [PubMed] [Google Scholar]
  31. C. Japhet, F. Nataf and F. Rogier, The optimized order 2 method : Application to convection-diffusion problems. Future Gener. Comp. Syst. 18 (2001) 17–30. [CrossRef] [Google Scholar]
  32. S. Linge, J. Sundnes, M. Hanslien, G.T. Lines and A. Tveito, Numerical solution of the bidomain equations. Phil. Trans. R. Soc. A. 367 (2009) 1931–1950. [CrossRef] [Google Scholar]
  33. G.T. Lines, M.L. Buist, P. Grottum, A.J. Pullan, J. Sundnes and A. Tveito, Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Vis. Sci. 5 (2003) 215–239. [CrossRef] [Google Scholar]
  34. P.-L. Lions, On the Schwarz alternating method. III : a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, edited by T.F. Chan, R. Glowinski, J. Périaux and O. Widlund, SIAM Philadelphia, PA (1990). [Google Scholar]
  35. L. Luo and Y. Rudy, A model of the ventricular cardiac action potential : depolarization, repolarization and their interaction. Circ. Res. 68 (1991) 1501–1526. [CrossRef] [PubMed] [Google Scholar]
  36. L. Mirabella, F. Nobile and A. Veneziani, An a posteriori error estimator for model adaptivity in electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 2727–2737. [CrossRef] [Google Scholar]
  37. M. Munteanu, L.F. Pavarino and S. Scacchi, A scalable Newton–Krylov–Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 3 (2009) 3861–3883. [CrossRef] [Google Scholar]
  38. F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm. M3AS 5 (1995) 67–93. [Google Scholar]
  39. L.F. Pavarino and S. Scacchi, Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420–443. [CrossRef] [Google Scholar]
  40. L.F. Pavarino and S. Scacchi, Parallel Multilevel Schwarz and block preconditioners for the Bidomain parabolic-parabolic and parabolic-elliptic formulations. SIAM J. Sci. Comput. 33 (2011) 1897–1919. [CrossRef] [Google Scholar]
  41. M. Pennacchio and V. Simoncini, Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49–70. [CrossRef] [MathSciNet] [Google Scholar]
  42. M. Pennacchio and V. Simoncini, Algebraic multigrid preconditioners for the Bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033–3050. [CrossRef] [Google Scholar]
  43. M. Pennacchio and V. Simoncini, Non-symmetric Algebraic Multigrid Preconditioners for the Bidomain reaction-diffusion system, in Numerical Mathematics and Advanced Applications, ENUMATH 2009, Part 2 (2010) 729–736. [Google Scholar]
  44. M. Perego and A. Veneziani, An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. ETNA 35 (2009) 234–256. [Google Scholar]
  45. M. Potse, B. Dubé, J. Richer and A. Vinet, A comparison of Monodomain and Bidomain Reaction-Diffusion models for Action Potential Propagation in the Human Heart. IEEE Trans. Biomed. Eng. 53 (2006) 2425–2435,. [CrossRef] [PubMed] [Google Scholar]
  46. A.J. Pullan, M.L. Buist and L.K. Cheng, Mathematical Modelling the Electrical Activity of the Heart. World Scientific, Singapore (2005). [Google Scholar]
  47. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). [Google Scholar]
  48. B.J. Roth, Action potential propagation in a thick strand of cardiac muscle. Circ. Res. 68 (1991) 162–173. [CrossRef] [PubMed] [Google Scholar]
  49. F.B. Sachse, Computational Cardiology. Springer, Berlin (2004). [Google Scholar]
  50. S. Scacchi, A hybrid multilevel Schwarz method for the Bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051–4061. [CrossRef] [Google Scholar]
  51. B.F. Smith, P.E. Bjørstad and W. Gropp. Domain Decomposition : Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996). [Google Scholar]
  52. 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) 233–248. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  53. A. Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86 (2000) 733–752. [CrossRef] [MathSciNet] [Google Scholar]
  54. A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory. Springer Ser. Comput. Math. 34 (2004). [Google Scholar]
  55. M. Veneroni, Reaction-diffusion systems for the macroscopic Bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10 (2009) 849–868. [CrossRef] [MathSciNet] [Google Scholar]
  56. E.J. Vigmond, F. Aguel and N.A. Trayanova, Computational techniques for solving the Bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 1260–1269. [CrossRef] [PubMed] [Google Scholar]
  57. E.J. Vigmond, R. Weber dos Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the caridac Bidomain equations. Prog. Biophys. Mol. Biol. 96 (2008) 3–18. [CrossRef] [PubMed] [Google Scholar]
  58. R. Weber dos Santos, G. Planck, S. Bauer and E.J. Vigmond, Parallel multigrid preconditioner for the cardiac Bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 1960–1968. [CrossRef] [PubMed] [Google Scholar]
  59. J. Xu and J. Zou, Some nonoverlapping domain decomposition methods. SIAM Review 40 (1998) 857–914. [CrossRef] [MathSciNet] [Google Scholar]
  60. J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Review 34 (1992) 581–613. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you