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Mathematics of Computation

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Polynomial extension operators. Part III

Authors: L. Demkowicz, J. Gopalakrishnan and J. Schöberl
Journal: Math. Comp. 81 (2012), 1289-1326
MSC (2010): Primary 46E35, 46E40; Secondary 41A10, 65D05, 65L60
Published electronically: September 20, 2011
MathSciNet review: 2904580
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Abstract: In this concluding part of a series of papers on tetrahedral polynomial extension operators, the existence of a polynomial extension operator in the Sobolev space $ \boldsymbol{H}({div})$ is proven constructively. Specifically, on any tetrahedron $ K$, given a function $ w$ on the boundary $ \partial K$ that is a polynomial on each face, the extension operator applied to $ w$ gives a vector function whose components are polynomials of at most the same degree in the tetrahedron. The vector function is an extension in the sense that the trace of its normal component on the boundary $ \partial K$ coincides with $ w$. Furthermore, the extension operator is continuous from $ H^{-1/2}(\partial K)$ into $ \boldsymbol{H}({div},K)$. The main application of this result and the results of this series of papers is the existence of commuting projectors with good $ hp$-approximation properties.

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Additional Information

L. Demkowicz
Affiliation: Institute of Computational Engineering and Sciences, 1 University Station, C0200, The University of Texas at Austin, Texas 78712

J. Gopalakrishnan
Affiliation: University of Florida, Department of Mathematics, Gainesville, Florida 32611–8105

J. Schöberl
Affiliation: Technische Universität Wein, Wiedner Hauptstrasse 8-10, Wein 1040, Austria

Received by editor(s): January 3, 2010
Received by editor(s) in revised form: February 23, 2011
Published electronically: September 20, 2011
Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-1014817, the Johann Radon Institute for Computational and Applied Mathematics (RICAM), and the FWF-Start-Project Y-192 “hp-FEM”
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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