An ε-uniform Ritz-Galerkin finite element method for numerical solution of singularly perturbed delay g differential equations

A boundary value problem for second order singularly perturbed delay differential equation is considered with the delay and advance arguments that are sufficiently small. Such problems have earlier been tackled asymptotically by the researchers Lange and Miura [9], [10]. The numerical treatment of the problem is given in Kadalbajoo and Sharma[4], [5], [6], they have used fitted mesh finite difference scheme and shown the order of convergence is one. In this paper, we have taken a piecewise-uniform fitted mesh (Shishkin mesh) to resolve the boundary layer and we have shown that Ritz-Galerkin method has almost second order parameters-uniform convergence. Several test examples are solved to demonstrate the efficiency of the method and how the size of the delay and advance arguments affect the layer behavior of the solution.