Uniformly Convergent Nite Element Methods for Singularly Perturbed Elliptic Boundary Value Problems: Convection-diiusion Type

In this paper we consider the standard bilinear nite element method (FEM) and the corresponding streamline diiusion FEM for the singularly perturbed elliptic boundary value problem ?" (@ 2 u @x 2 + @ 2 u @y 2) ? b(x; y) ru + a (x; y)u = f (x; y) in the two space dimensions. By using the asymptotic expansion method of Vishik and Lyusternik 42] and the technique we used in 25, 26], we prove that the standard bilinear FEM on a Shishkin type mesh achieves rst-order uniform convergence rate globally in L 2 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 diiusion FEM.