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Error estimates for a FitzHugh-Nagumo parameter-dependent reaction-diffusion system. (English) Zbl 1272.65072
The FitzHugh-Nagumo (FHN) system, which consists of two parabolic partial differential equations that are coupled through nonlinear terms were proposed for the modeling of the transmission of electrical impulses in a nerve axon (see, e.g. [R. FitzHugh, Biophys. J. 1, 445–466 (1961)]). A space-time approximation of the FHN system is studied here (cf. [K. Chrysafinos et al., ESAIM, Math. Model. Numer. Anal. 42, No. 1, 25–55 (2008; Zbl 1136.65089)]). The object of the authors is to derive stability and error estimates of arbitrary order for which time discretization and spatial discretization parameters can be chosen independently. Convergence under minimal regularity assumptions on the given data are demonstrated. Optimal error estimates are derived when the solutions are sufficiently smooth.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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