Global pointwise error estimates for uniformly convergent finite element methods for the elliptic boundary layer problem

Abstract

This paper continues our discussion for the anisotropic model problem −(ε 2 ∂ 2u ∂x 2 + ∂ 2 ∂y 2 ) + a(x,y)u = f(x,y) in [1]. There we constructed a bilinear finite element method on a Shishkin type mesh. The method was shown to be convergent, independent of the small parameter ϵ, in the order of N −2ln 2 N in the L 2-norm, where N 2 is the total number of mesh points. In this paper, the method is shown to be convergent, independent of ϵ, in the order of N −2ln 3 N in the L∞-norm in the whole computational domain, which explains the uniform convergence phenomena we found in the numerical results in [1]. Another numerical experiment is presented here, which confirms our theoretical analysis. Published by Elsevier Science Ltd.