ON THE CONVERGENCE BEHAVIOR OF ADAPTIVE FINITE ELEMENT METHODS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

Singularly perturbed boundary value problems remain a persistent challenge in numerical analysis because solutions typically exhibit thin boundary or interior layers whose width scales with a small perturbation parameter epsilon. Uniform meshes fail to resolve these layers without prohibitive refinement, motivating the use of adaptive finite element methods (AFEM) driven by a posteriori error estimators. This paper investigates the convergence behavior of a residual-based adaptive finite element scheme applied to a family of one- and two-dimensional reaction-diffusion problems with boundary layers. A sequence of adaptively refined meshes was generated using a Dorfler marking strategy combined with newest-vertex bisection, and the resulting error decay was measured in the energy norm across a range of perturbation parameters spanning epsilon = 10^-2 to 10^-6. Five sets of computational data are reported, covering degrees-of-freedom versus error decay, effectivity indices of the estimator, layer-resolution ratios, computational cost comparisons against uniform refinement, and parameter-robustness trends. The results indicate that the adaptive scheme recovers close to optimal convergence order regardless of epsilon, whereas uniform refinement degrades sharply as epsilon decreases. The estimator-maintained effectivity indices between 0.85 and 1.30 across all tested regimes, confirming reliability and efficiency in the sense of a posteriori error control.