Error estimates for the semidiscrete Galerkin approximations of the FitzHugh-Nagumo equations
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 \textit{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.
transmission of electrical impulses in a nerve axon
scientific article; zbMATH DE number 64325
scientific article