Approximation of singularly perturbed reaction-diffusion equations on adaptive meshes

The Dirichlet problem for a parabolic reaction-diffusion equation is considered on a segment. The highest derivative of the equation is multiplied by a parameter $\varepsilon$ taking arbitrary values in the half-interval (0,1]. For this problem we consider classical difference approximations of the equations on sequentially locally refined (a posteriori) meshes. On the subdomains subjected to refining, which are defined by the gradients of the mesh solutions of the intermediate problems, uniform meshes are used. We construct special schemes which allow us to obtain the approximations that converge "almost $\varepsilon$-uniformly",i.e., with an error weakly depending on $\varepsilon\colon |u(x,t)-z(x,t)|\le M[N_1^{-2/3}+\varepsilon ^{-\nu}N_1^{-1}+N_0^{-1}]$, $(x,t)\in\overline G_h$, where $\nu$ is an arbitrary number from (0,1]; $N_1+1$ and $N_0+1$ are the number of the mesh nodes in $x$ and $t$.