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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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