Stable broken $H^1$ and $\mathbfit{H}(\mathrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions
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- Math. Comp. 89 (2020), 551-594 Request permission
Abstract:
We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the $H^1$ setting, we look for functions whose jumps across the faces are prescribed, whereas in the $\mathbfit {H}(\mathrm {div})$ setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken $H^1$ and $\mathbfit {H}(\mathrm {div})$ spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.References
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Additional Information
- Alexandre Ern
- Affiliation: Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France; and Inria, 2 rue Simone Iff, 75589 Paris, France
- MR Author ID: 349433
- Email: alexandre.ern@enpc.fr
- Martin Vohralík
- Affiliation: Inria, 2 rue Simone Iff, 75589 Paris, France; and Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France
- ORCID: 0000-0002-8838-7689
- Email: martin.vohralik@inria.fr
- Received by editor(s): December 23, 2016
- Received by editor(s) in revised form: August 19, 2018
- Published electronically: October 4, 2019
- Additional Notes: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR)
The second author is the corresponding author. - © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 551-594
- MSC (2010): Primary 65N15, 65N30, 76M10
- DOI: https://doi.org/10.1090/mcom/3482
- MathSciNet review: 4044442