Analysis of the SDFEM for singularly perturbed differential-difference equations

In this paper, the stability and accuracy of a streamline diffusion finite element method (SDFEM) for the singularly perturbed differential-difference equation of convection term with a small shift is considered. With a special choice of the stabilization quadratic bubble function and by using the discrete Green's function, the new method is shown to have an optimal second order in the sense that , where is the SDFEM approximation of the exact solution u in linear finite element space . At last, a second order uniform convergence result for the SDFEM is obtained. Numerical results are given to confirm the -uniform convergence rate of the nodal errors.