Abstract: Singularly perturbed elliptic equations generate multiscale solution profiles containing smooth outer regions and narrow boundary or interior layers. Uniform finite element meshes often waste degrees of freedom away from layers while under-resolving the singular regions. The present article develops an adaptive finite element framework for reaction-diffusion type singular perturbation problems, with residual based a posteriori indicators used to drive local mesh refinement. The analysis is presented in the energy norm associated with the perturbed bilinear form. The paper formulates the weak problem in Sobolev spaces, states the Galerkin finite element approximation, derives local residual and flux-jump indicators, and establishes reliability, local efficiency, convergence, and computational complexity statements in a standard theorem-proof structure. A reproducible one-dimensional benchmark problem with known exact solution is used to compare uniform and adaptive refinement. The numerical results show that adaptive refinement concentrates nodes near the boundary layers, reduces the error with fewer elements, and produces a practical effectivity index for estimator guided refinement. The article includes implementation details, algorithmic pseudocode, convergence tables, mesh plots, residual indicator visualizations, and computational cost comparisons. The resulting framework provides a publication-ready model for adaptive computation of multiscale singularly perturbed elliptic problems.
Hegde et al. (Sun,) studied this question.