Error estimates for the semidiscrete Galerkin approximations of the FitzHugh-Nagumo equations
type: Item
instance of: scholarly article
title: Error estimates for the semidiscrete Galerkin approximations of the FitzHugh-Nagumo equations
zbMATH Open document ID: 0757.65125
DOI: 10.1016/0096-3003(92)90013-Q
author: S. Singh
published in: Applied Mathematics and Computation
publication date: 1992-09-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 35Q80
zbMATH DE Number: 64325
zbMATH Keywords: semidiscrete Galerkin approximation
zbMATH Keywords: Fitzhugh-Nagumo equations
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: transmission of electrical impulses in a nerve axon
zbMATH Keywords: error estimates
MaRDI profile type: Publication
cites work: Q4073917
cites work: Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order
cites work: Some Mathematical Problems from Neurobiology
cites work: Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data
cites work: Geometric theory of semilinear parabolic equations
cites work: Finite Element Approximation of the Nonstationary Navier–Stokes Problem III. Smoothing Property and Higher Order Error Estimates for Spatial Discretization
cites work: Existence and regularity for the Fitzhugh-Nagumo equations with inhomogeneous boundary conditions
cites work: Convergence of Successive Iterative Semidiscretizations for Fitzhugh–Nagumo Reaction Diffusion Systems
cites work: Error Estimates for Spatially Discrete Approximations of Semilinear Parabolic Equations with Nonsmooth Initial Data
cites work: Convergence Estimates for Semidiscrete Approximations of Nonselfadjoint Parabolic Equations
cites work: Q3326968
cites work: Q5634306
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4095249
cites work: Q4108677
cites work: Qualitative theory of the Fitz Hugh-Nagumo equations
cites work: Boundary value problems for the Fitzhugh-Nagumo equations
cites work: Galerkin finite element methods for parabolic problems
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description: scientific article; zbMATH DE number 64325