Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection—diffusion type Original Research Article

In this paper we consider the standard bilinear finite element method (FEM) and the corresponding streamline diffusion FEM for the singularly perturbed elliptic boundary value problem −εα(∂2u∂x2 + ∂2u∂y2) − b(x, y) · ▽u + aα(x, y)u = f(x, y) in the two space dimensions. By using the asymptotic expansion method of Vishik and Lyusternik [36] and the technique we used in [21,22], we prove that the standard bilinear FEM on a Shishkin type mesh achieves first-order uniform convergence rate globally in L2 norm for both the ordinary exponential boundary layer case and the parabolic boundary layer case. Extensive numerical results are carried out for both cases. The results show that our methods perform much better than either the classical standard or streamline diffusion FEM.