We consider a family of singularly perturbed elliptic problems in two dimensions. A novel fitted operator finite difference method developed is proposed to solve this problems. Through a rigorous convergence analysis, we show that the method is second order convergent in both variables. Further attempts are made to improve the order of convergence via some convergence acceleration techniques, namely the Richardson extrapolation. In turn, we achieve fourth order accurate results. Error analysis after extrapolation is also presented. Furthermore, some numerical results confirming the theoretical estimates are provided. We also compare our results with those obtained in the literature (see, e.g., [R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal. 47(1) 89--108.] and noticed that the error obtained by our approach is exponentially smaller than the one obtained by their approach.
['Justin B. Munyakazi', 'Kailash C. Patidar']
Higher order numerical methods for singularly perturbed elliptic problems