Improvement of the Fitted Mesh Method by Three Transition Points Technique for Singularly Perturbed Problem with Two Parameters

Singularly perturbed ordinary differential equation with two small parameters is considered. Firstly, the solution is decomposed into the smooth component and the singular component. The upper bounds of the smooth component and the singular component are studied. Secondly, the traditional Shishkin's scheme is presented and it is proved to be uniformly convergent with respect to the small parameter. Thirdly the technique of three transition points is introduced in order to improve the order of convergence. Three transition points scheme captures the property of boundary layer very well. It is a non equidistant method. It is proved to be uniformly convergent with respect to the small parameter in order one, which is higher than the traditional Shishkin's scheme. Finally, numerical results are given, which are in agreement with the theoretical results.