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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.

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)
Full Text: DOI
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