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Error estimates for the semidiscrete Galerkin approximations of the FitzHugh-Nagumo equations. (English) Zbl 0757.65125
The author obtains $$L^ 2$$ error estimates for a semidiscrete Galerkin approximation of the Fitzhugh-Nagumo equations with nonsmooth initial data. The equations consist of a nonlinear parabolic equation coupled with an ordinary differential equation which arise in the study of transmission of electrical impulses in a nerve axon and have the form $$u_ t=\nabla^ 2 u+F(u)-v$$, $$v_ t=\sigma u-\gamma v$$ on $$Q=\Omega\times(0,T)$$ where $$F(u)=u(1-u)(u-a)$$, with the initial conditions $$u(x,0)=u_ 0(x)$$, $$v(x,0)=v_ 0(x)$$, and the boundary condition for $$u$$ $$u(x,t)|_{\partial \Omega}=0$$.
The error estimates obtained are of the form $$\| u(t)-u_ h(t)\|\leq c\cdot h^ \mu t^{(-y-\sigma)/2}$$, $$t>0$$, $$\| v(t)- v_ h(t)\|\leq c\cdot h^ \mu$$ where $$\mu$$ and $$\sigma$$ depend on $$\gamma$$, the regularity of $$u_ 0(x)$$ and $$v_ 0(x)$$, and on compatibility conditions.
Earlier J. W. Jerome [SIAM J. Numer. Anal. 17, 192-206 (1980; Zbl 0434.65095)] used a discrete approximation in time to obtain the existence of a global solution for the Fitzhugh-Nagumo equations but did not give error estimates.
In the present paper use is made of holomorphic semigroups $$S(t)=e^{- At}$$ and $$S_ h(t)=e^{-A_ ht}$$ generated infinitesimally by the operators $$-A=\nabla^ 2$$ and $$-A_ h$$ (approximation of operator $$A$$), respectively, to obtain the error estimates. Smoothing due to the parabolic equation is used to show increased accuracy for the approximation for positive time.

##### MSC:
 65Z05 Applications to the sciences 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 92B20 Neural networks for/in biological studies, artificial life and related topics 35Q80 Applications of PDE in areas other than physics (MSC2000)
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