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

Leykekhman, Dmitriy

doi:10.1090/S0025-5718-07-02015-7

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

Math. Comp. 77 (2008), 21-39 Request permission

Abstract:

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.

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