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Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations


Author: Natalia Kopteva
Journal: Math. Comp. 83 (2014), 2061-2070
MSC (2010): Primary 65N15, 65N30, 65N50; Secondary 65N06
DOI: https://doi.org/10.1090/S0025-5718-2014-02820-2
Published electronically: February 28, 2014
MathSciNet review: 3223324
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Abstract: We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp
boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.


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Additional Information

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
Address at time of publication: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XT, United Kingdom
Email: natalia.kopteva@strath.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2014-02820-2
Keywords: Anisotropic triangulation, linear finite elements, maximum norm, singular perturbation, Bakhvalov mesh, Shishkin mesh
Received by editor(s): September 14, 2012
Received by editor(s) in revised form: January 22, 2013
Published electronically: February 28, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.