Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion Type

Abstract

We consider the bilinear finite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem −ϵ 2( ι 2u ιx 2 + ι 2u ιy 2 ) + a(x,y)u = f(x,y) in two space dimensions. By using a very sophisticated asymptotic expansion of Han et al. [1] and the technique we used in [2], we prove that our method achieves almost second-order uniform convergence rate in L 2-norm. Numerical results confirm our theoretical analysis.