Resolvent estimates for boundary value problems on large intervals via the theory of discrete approximations

In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic - parabolic equations. Furthermore, we show that our results apply to the FitzHugh - Nagumo system.