Conditional $\varepsilon$-uniform convergence of adaptation algorithms in the finite element method for singularly perturbed problems

The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval $[-1, 1]$ is examined. The highest derivative in this equation appears with a small parameter $\varepsilon^2$ ($\varepsilon\in (0, 1]$). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints. An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process, the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown to converge "conditionally $\varepsilon$-uniformly" to some limit partition for which the error estimate $O(N^{-2}\ln^3N)$ is proved. The main results are obtained under the assumption that $\varepsilon\ll N^{-1}$, where $N$ is number of grid nodes; thus, conditional $\varepsilon$-uniform convergence is dealt with. The proofs use the Galerkin projector and its property to be quasi-optimal.