Reciprocal polynomial extrapolation vs Richardson extrapolation for singular perturbed boundary problems

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1–10, 2004), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ϵ. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed.