Abstract
AbstractIn this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Equ 28:955, 2016). For small spatial grid-sizes, we establish some useful Fredholm properties for the operator that arises after linearizing our system around the travelling wave solutions to the original Nagumo PDE. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is a spatially stretched and twisted version of the standard second order differential operator that is associated to the PDE waves.
Funder
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
National Science Foundation
Publisher
Springer Science and Business Media LLC
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