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Error estimates on anisotropic ${\mathcal Q}_1$ elements for functions in weighted Sobolev spaces


Authors: Ricardo G. Durán and Ariel L. Lombardi
Journal: Math. Comp. 74 (2005), 1679-1706
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-05-01732-1
Published electronically: March 29, 2005
MathSciNet review: 2164092
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Abstract: In this paper we prove error estimates for a piecewise $\mathcal{Q}_1$average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions.

Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses.

Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems.

Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements.

As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.


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Additional Information

Ricardo G. Durán
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: rduran@dm.uba.ar

Ariel L. Lombardi
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: aldoc7@dm.uba.ar

DOI: https://doi.org/10.1090/S0025-5718-05-01732-1
Keywords: Anisotropic elements, weighted norms
Received by editor(s): August 4, 2003
Received by editor(s) in revised form: January 8, 2004
Published electronically: March 29, 2005
Additional Notes: The research was supported by ANPCyT under grant PICT 03-05009 and by CONICET under grant PIP 0660/98. The first author is a member of CONICET, Argentina.
Article copyright: © Copyright 2005 American Mathematical Society

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