Free Access
Volume 47, Number 1, January-February 2013
Page(s) 281 - 304
Published online 23 November 2012
  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 : M2AN 38 (2004) 261–289. [CrossRef] [EDP Sciences]
  3. G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comput. 67 (1998) 457–477. [CrossRef] [MathSciNet]
  4. C. Chiu and N.J. Walkington, An ADI method for hysteric reaction-diffusion systems. SIAM J. Numer. Anal. 34 (1997) 1185–1206. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  6. K. Chrysafinos and N.J. Walkington, Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM : M2AN 42 (2008) 27–56. [CrossRef] [EDP Sciences]
  7. K. Chrysafinos and N.J. Walkington, Discontinous Galerkin approximations of the Stokes and Navier–Stokes problem. Math. Comput. 79 (2010) 2135–2167. [CrossRef]
  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. [CrossRef]
  10. S, Descombes and M. Ribot, Convergence of the Peaceman–Rachford approximation for reaction-diffusion systems. Numer. Math. 95 (2003) 503–525. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  13. K. Ericksson and C. Johnson, Adaptive finite element methods for parabolic problems IV : Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. [CrossRef] [MathSciNet]
  14. K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. ESAIM : M2AN 29 (1985) 611–643.
  15. D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic equations. ESAIM : M2AN 27 (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. [CrossRef] [PubMed]
  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. [CrossRef] [MathSciNet]
  21. M.R. Garvie and J.M. Blowey, A reaction-diffusion system of λω type. Part II : Numerical analysis. Eur. J. Appl. Math. 16 (2005) 621–646. [CrossRef]
  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. [CrossRef]
  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. [CrossRef]
  25. E. Hansen and A. Ostermann, Dimension splitting for evolution equations. Numer. Math. 108 (2008) 557–570. [CrossRef] [MathSciNet]
  26. S.P. Hastings, Some mathematical models from neurobiology. Amer. Math. Monthly 82 (1975) 881–895. [CrossRef] [MathSciNet]
  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. [CrossRef]
  29. D. Jackson, Error estimates for the semidiscrete Galerkin approximations of the FitzHugh–Nagumo equations. Appl. Math. Comput. 50 (1992) 93–114. [CrossRef]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  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. [CrossRef]
  35. J.S. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proc. IRE 50 (1962) 2061–2070. [CrossRef]
  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. [CrossRef]
  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. USA 97 (2000) 9840–9843. [CrossRef]
  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. [CrossRef] [MathSciNet]
  42. E. Zeidler, Nonlinear functional analysis and its applications, in II/B Nonlinear monotone operators. Springer-Verlag, New York (1990).

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