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Finite element approximations in a non-Lipschitz domain: Part II


Authors: Gabriel Acosta and María G. Armentano
Journal: Math. Comp. 80 (2011), 1949-1978
MSC (2010): Primary 65N30, 46E35
Published electronically: April 1, 2011
MathSciNet review: 2813345
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Abstract: In a paper by R. Durán, A. Lombardi, and the authors (2007) the finite element method was applied to a non-homogeneous Neumann problem on a cuspidal domain $ \Omega\subset \mathbb{R}^2$, and quasi-optimal order error estimates in the energy norm were obtained for certain graded meshes. In this paper, we study the error in the $ L^2$ norm obtaining similar results by using graded meshes of the type considered in that paper. Since many classical results in the theory Sobolev spaces do not apply to the domain under consideration, our estimates require a particular duality treatment working on appropriate weighted spaces.

On the other hand, since the discrete domain $ \Omega_h$ verifies $ \Omega\subset \Omega_h$, in the above-mentioned paper the source term of the Poisson problem was taken equal to 0 outside $ \Omega$ in the variational discrete formulation. In this article we also consider the case in which this condition does not hold and obtain more general estimates, which can be useful in different problems, for instance in the study of the effect of numerical integration, or in eigenvalue approximations.


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Additional Information

Gabriel Acosta
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: gacosta@dm.uba.ar

María G. Armentano
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: garmenta@dm.uba.ar

DOI: https://doi.org/10.1090/S0025-5718-2011-02481-6
Keywords: Cuspidal domains, finite elements, graded meshes
Received by editor(s): April 6, 2009
Received by editor(s) in revised form: September 13, 2010
Published electronically: April 1, 2011
Additional Notes: This work was supported by ANPCyT under grants PICT 2006-01307 and PICT-2007-00910, and by the Universidad de Buenos Aires under grant X007 and by CONICET under grant PIP 5478/1438
Article copyright: © Copyright 2011 American Mathematical Society