Analysis of the SDFEM for singularly perturbed differential–difference equations

Abstract

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 $$\Vert u-u_{h}\Vert _{\infty }\le C\inf \nolimits _{v_h\in V^h}\Vert u-v_{h}\Vert _{\infty }$$ , where $$u_{h}$$ is the SDFEM approximation of the exact solution u in linear finite element space $$V_{h}$$ . At last, a second order uniform convergence result for the SDFEM is obtained. Numerical results are given to confirm the $$\varepsilon $$ -uniform convergence rate of the nodal errors.