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


Author: Dmitriy Leykekhman
Journal: Math. Comp. 77 (2008), 21-39
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-07-02015-7
Published electronically: May 14, 2007
MathSciNet review: 2353942
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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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Additional Information

Dmitriy Leykekhman
Affiliation: Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005
Email: dmitriy@caam.rice.edu

DOI: https://doi.org/10.1090/S0025-5718-07-02015-7
Keywords: Finite element, singularly perturbed, pointwise estimates, reaction-diffusion
Received by editor(s): June 8, 2005
Received by editor(s) in revised form: November 18, 2006
Published electronically: May 14, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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