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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

   

 

Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra


Authors: Gabriel Acosta, Thomas Apel, Ricardo G. Durán and Ariel L. Lombardi
Journal: Math. Comp. 80 (2011), 141-163
MSC (2010): Primary 65N30
Published electronically: July 29, 2010
MathSciNet review: 2728975
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Abstract: We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra.

Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree $ k\ge 0$, we prove error estimates of order $ j+1$ when the vector field being approximated has components in $ W^{j+1,p}$, for triangles or tetrahedra, where $ 0\le j \le k$ and $ 1\le p \le\infty$. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case $ j=k$. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for $ k=0$.


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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. Member of CONICET, Argentina.
Email: gacosta@dm.uba.ar

Thomas Apel
Affiliation: Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Neubiberg, Germany.
Email: thomas.apel@unibw.de

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

Ariel L. Lombardi
Affiliation: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, B1613GSX Provincia de Buenos Aires, Argentina and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. Member of CONICET, Argentina.
Email: aldoc7@dm.uba.ar

DOI: http://dx.doi.org/10.1090/S0025-5718-2010-02406-8
Keywords: Mixed finite elements, Raviart-Thomas, anisotropic finite elements
Received by editor(s): September 11, 2008
Received by editor(s) in revised form: May 2, 2009
Published electronically: July 29, 2010
Additional Notes: The work of the first author was supported by Deutsche Forschungsgemeinschaft, Grant AP 72/3-1 and by Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), Argentina, Grant PAV–120
The first, third and fourth authors were partially supported by ANPCyT, under grants PICT 2007-910, PICT 2005-33617, and PICT 2007-01307, and by Universidad de Buenos Aires, under Grant X070.
Article copyright: © Copyright 2010 American Mathematical Society



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