Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

Konstantinos Chrysafinos; Sotirios P. Filopoulos; Theodosios K. Papathanasiou

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 47, Issue: 1, page 281-304
  • ISSN: 0764-583X

Abstract

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Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present. Various physical parameters appearing in the model are tracked and numerical examples are presented.

How to cite

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Chrysafinos, Konstantinos, Filopoulos, Sotirios P., and Papathanasiou, Theodosios K.. "Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2012): 281-304. <http://eudml.org/doc/244360>.

@article{Chrysafinos2012,
abstract = {Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present. Various physical parameters appearing in the model are tracked and numerical examples are presented.},
author = {Chrysafinos, Konstantinos, Filopoulos, Sotirios P., Papathanasiou, Theodosios K.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Error estimates; discontinuous time-stepping Galerkin schemes; FitzHugh–Nagumo equations; reaction-diffusion; parameter dependent; coarse time-stepping; FitzHugh-Nagumo equations; reaction-diffusion system; error estimates; Galerkin schemes; discontinuous time-stepping; stability; convergence},
language = {eng},
month = {11},
number = {1},
pages = {281-304},
publisher = {EDP Sciences},
title = {Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system},
url = {http://eudml.org/doc/244360},
volume = {47},
year = {2012},
}

TY - JOUR
AU - Chrysafinos, Konstantinos
AU - Filopoulos, Sotirios P.
AU - Papathanasiou, Theodosios K.
TI - Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/11//
PB - EDP Sciences
VL - 47
IS - 1
SP - 281
EP - 304
AB - Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present. Various physical parameters appearing in the model are tracked and numerical examples are presented.
LA - eng
KW - Error estimates; discontinuous time-stepping Galerkin schemes; FitzHugh–Nagumo equations; reaction-diffusion; parameter dependent; coarse time-stepping; FitzHugh-Nagumo equations; reaction-diffusion system; error estimates; Galerkin schemes; discontinuous time-stepping; stability; convergence
UR - http://eudml.org/doc/244360
ER -

References

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  1. G. Akrivis and M. Crouzeix, Linearly implicit methods for nonlinear parabolic equations. Math. Comput.73 (2004) 613–635.  
  2. G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM : M2AN38 (2004) 261–289.  
  3. G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comput.67 (1998) 457–477.  
  4. C. Chiu and N.J. Walkington, An ADI method for hysteric reaction-diffusion systems. SIAM J. Numer. Anal.34 (1997) 1185–1206.  
  5. K. Chrysafinos and N.J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal.44 (2006) 349–366.  
  6. K. Chrysafinos and N.J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM : M2AN42 (2008) 27–56.  
  7. K. Chrysafinos and N.J. Walkington, Discontinous Galerkin approximations of the Stokes and Navier–Stokes problem. Math. Comput.79 (2010) 2135–2167.  
  8. P.G. Ciarlet, The finite element method for elliptic problems. SIAM Classics Appl. Math. (2002).  
  9. M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations. Math. Comput.36 (1981) 455–473.  
  10. S, Descombes and M. Ribot, Convergence of the Peaceman–Rachford approximation for reaction-diffusion systems. Numer. Math.95 (2003) 503–525.  
  11. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal.28 (1991) 43–77.  
  12. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in L∞(L2) and L∞(L∞). SIAM J. Numer. Anal.32 (1995) 706–740.  
  13. K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV : Nonlinear problems. SIAM J. Numer. Anal.32 (1995) 1729–1749.  
  14. K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. ESAIM : M2AN29 (1985) 611–643.  
  15. D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic equations. ESAIM : M2AN27 (1993) 35–54.  
  16. D. Estep, M. Larson and R. Williams, Estimating the error of numerical solutions of systems of reaction-diffusion equations. Mem. Amer. Math. Soc.146 (2000) viii+109.  
  17. L. Evans, Partial Differential Equations. AMS, Providence, RI (1998).  
  18. P. Fife, Mathematical aspects of reacting and diffusing systems. Lect. Notes Biomath.28 (1978).  
  19. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J.1 (1961) 445–466.  
  20. P. Franzone, P. Deflhard, B. Erdmann, J. Lang and L. Pavarino, Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput.28 (2006) 942–962.  
  21. M.R. Garvie and J.M. Blowey, A reaction-diffusion system of λ − ω type. Part II : Numerical analysis. Eur. J. Appl. Math.16 (2005) 621–646.  
  22. M.R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator interactions with the Holling type II functional response. Numer. Math.107 (2008) 641–667.  
  23. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes. Springer-Verlag, New York (1986).  
  24. M.D. Gunzburger, L.S. Hou and W. Zhu, Fully discrete finite element approximation of the forced Fisher equation. J. Math. Anal. Appl.313 (2006) 419–440.  
  25. E. Hansen and A. Ostermann, Dimension splitting for evolution equations. Numer. Math.108 (2008) 557–570.  
  26. S.P. Hastings, Some mathematical models from neurobiology. Amer. Math. Monthly82 (1975) 881–895.  
  27. W. Hundsdorfer and J. Verwer, Numerical solution for time-dependent advection-diffusion-reaction equations. Springer-Verlag, Berlin (2003).  
  28. D. Jackson, Existence and regularity for the FitzHugh–Nagumo equations with inhomogeneous boundary conditions. Nonlinear Anal. Theory Methods Appl.14 (1990) 201–216.  
  29. D. Jackson, Error estimates for the semidiscrete Galerkin approximations of the FitzHugh–Nagumo equations. Appl. Math. Comput.50 (1992) 93–114.  
  30. P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal.15 (1978) 912–928.  
  31. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge (1987).  
  32. P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical aspects of finite elements in partial differential equations, edited by C. de Boor. Academic Press, New York (1974) 89–123.  
  33. D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I : Problems without control constraints. SIAM J. Control. Optim.47 (2008) 1150–1177.  
  34. C. Nagaiah, K. Kunisch and G. Plank, Numerical solution for optimal control problems of the reaction diffusion equations in cardiac electrophysiology. Comput. Optim. Appl.49 (2011) 149–178.  
  35. J.S. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proc. IRE50 (1962) 2061–2070.  
  36. C.S. Peskin, Partial Differential Equations in Biology. Courant Institute of Mathematical Sciences, New York (1975).  
  37. M.E. Schoenbek, Boundary value problems for the FitzHugh–Nagumo equations. J. Differ. Equ.30 (1978) 119–147.  
  38. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci.68 (1997).  
  39. C. Theodoropoulos, Y.-H. Qian and I.G. Kevrekidis, “Coarse” stability and bifurcation analysis using time-steppers : a reaction-diffusion example. Proc. Natl. Acad. Sci. USA97 (2000) 9840–9843.  
  40. V. Thomée, Galerkin finite element methods for parabolic problems. Spinger-Verlag, Berlin (1997).  
  41. N.J. Walkington, Compactness properties of CG and DG schemes. SIAM J. Numer. Anal.47 (2010) 4680–4710.  
  42. E. Zeidler, Nonlinear functional analysis and its applications, in II/B Nonlinear monotone operators. Springer-Verlag, New York (1990).  

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