Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Consider the problem $-\epsilon^2\Delta u+u=f$ with homogeneous Neumann boundary condition in a bounded smooth domain in $\mathbb{R}^N$. The whole range $0<\epsilon\le 1$ is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size $h$; the mesh is fixed and independent of $\epsilon$.

A precise analysis of how the error at each point depends on $h$ and $\epsilon$ is presented. As an application, first order error estimates in $h$, which are uniform with respect to $\epsilon$, are given.