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On commuting $ p$-version projection-based interpolation on tetrahedra


Authors: J. M. Melenk and C. Rojik
Journal: Math. Comp. 89 (2020), 45-87
MSC (2010): Primary 65N30; Secondary 65N35
DOI: https://doi.org/10.1090/mcom/3454
Published electronically: June 11, 2019
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Abstract: On the reference tetrahedron $ \widehat K$, we define three projection-based interpolation operators on $ H^2(\widehat K)$, $ {\mathbf H}^1(\widehat K,\operatorname {\mathbf {curl}})$, and $ {\mathbf H}^1(\widehat K,\operatorname {div})$. These operators are projections onto spaces of polynomials, they have the commuting diagram property, and they feature the optimal convergence rate as the polynomial degree increases in $ H^{1-s}(\widehat K)$, $ \widetilde {\mathbf {H}}^{-s}(\widehat K,\operatorname {\mathbf {curl}})$, and $ \widetilde {\mathbf {H}}^{-s}(\widehat K,\operatorname {div})$ for $ 0 \leq s \leq 1$.


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Additional Information

J. M. Melenk
Affiliation: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
Email: melenk@tuwien.ac.at

C. Rojik
Affiliation: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
Email: claudio.rojik@tuwien.ac.at

DOI: https://doi.org/10.1090/mcom/3454
Keywords: $p$-version FEM, commuting diagram, Maxwell equations
Received by editor(s): February 2, 2018
Received by editor(s) in revised form: October 12, 2018, and February 26, 2019
Published electronically: June 11, 2019
Additional Notes: The second author acknowledges the support of the Austrian Science Fund (FWF) under grant P 28367-N35
Article copyright: © Copyright 2019 American Mathematical Society