A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations

The researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed problem faced the question due to the loss of order of convergence, most of them justified it by quoting the simplicity of the method and there are some non-linear problems for which a parameter uniform scheme cannot be constructed based on fitted operator approach while for the same problem, a parameter uniform scheme is constructed based on fitted mesh method. Now question remains unanswered in the case of linear problem. In this article, we replied to this question by giving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operator approach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been constructed [M.K. Kadalbajoo, K.K. Sharma, ε uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49-62]. A theoretical reason behind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed. In support of the predicted theory, a number of numerical experiments are carried out.