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Mathematics of Computation

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Polynomial extension operators for $ H^1$, $ \boldsymbol{\mathit{H}}(\mathbf{curl})$ and $ \boldsymbol{\mathit{H}}(\mathbf{div})$-spaces on a cube


Authors: M. Costabel, M. Dauge and L. Demkowicz
Journal: Math. Comp. 77 (2008), 1967-1999
MSC (2000): Primary 65N35, 65N30; Secondary 78M10, 35J05
DOI: https://doi.org/10.1090/S0025-5718-08-02108-X
Published electronically: April 29, 2008
MathSciNet review: 2429871
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Abstract: This paper is devoted to the construction of continuous trace lifting operators compatible with the de Rham complex on the reference hexahedral element (the unit cube). We consider three trace operators: The standard one from $ H^1$, the tangential trace from $ \boldsymbol{\mathit{H}}(\mathbf{curl})$ and the normal trace from $ \boldsymbol{\mathit{H}}(\mathrm{div})$. For each of them we construct a continuous right inverse by separation of variables. More importantly, we consider the same trace operators acting from the polynomial spaces forming the exact sequence corresponding to the Nédélec hexahedron of the first type of degree $ p$. The core of the paper is the construction of polynomial trace liftings with operator norms bounded independently of the polynomial degree $ p$. This construction relies on a spectral decomposition of the trace data using discrete Dirichlet and Neumann eigenvectors on the unit interval, in combination with a result on interpolation between Sobolev norms in spaces of polynomials.


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Additional Information

M. Costabel
Affiliation: IRMAR, Université de Rennes 1, 35042 Rennes, France
Email: martin.costabel@univ-rennes1.fr

M. Dauge
Affiliation: IRMAR, Université de Rennes 1, 35042 Rennes, France
Email: monique.dauge@univ-rennes1.fr

L. Demkowicz
Affiliation: Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: leszek@ices.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02108-X
Keywords: Trace lifting, polynomial extension, de Rham complex, separation of variables
Received by editor(s): March 21, 2007
Received by editor(s) in revised form: September 6, 2007
Published electronically: April 29, 2008
Additional Notes: The work of the third author was supported in part by the Air Force under Contract F49620-98-1-0255.
Article copyright: © Copyright 2008 American Mathematical Society