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

J. Li, I.M. Navon

We consider the bilinear finite element method on a Shishkin mesh for the singularly perturbed elliptic boundary value problem $-\va^2({\pa^2u \over \pa x^2} + {\pa^2u \over \pa y^2})+a(x,y)u = f(x,y)$ in two space dimensions. By using a very sophisticated asymptotic expansion of Han {\it et al.} [11] and the technique we used in [17], we prove that our method achieves almost second-order uniform convergence rate in $L^2$-norm. Numerical results confirm our theoretical analysis.