In 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.

在本文中，我们考虑了Nagumo PDE的自适应空间离散化方案。该方案是一种基于均匀分布所考虑解决方案的弧长的常用空间网格自适应方法。我们假设这种公平分配是严格执行的，这导致了我们在Hupkes和Van Vleck中得出的无限范围相互作用的非局部问题（J Dyn Differ Equ 28：955，2016）。对于较小的空间网格大小，我们为操作员建立了一些有用的Fredholm属性，这些属性是在围绕原始Nagumo PDE的行波解线性化我们的系统之后产生的。特别是，我们执行奇异摄动参数，以从自然极限算符中提升这些属性。