Quasioptimal uniformly convergent finite element methods for the elliptic boundary layer problem

We consider the bilinear finite element method for the singularly perturbed elliptic boundary value problem in two dimensions. We construct our method on a Shishkin mesh. Through the boundary layer correction technique and elaborate barrier functions, the method is shown to be convergent, uniformly in the perturbation parameter of order N−2 ln2 N in the L2 norm. Numerical results confirm our theoretical analysis.