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Mathematics of Computation

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Stable broken $ H^1$ and $ H(\mathrm{div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

Authors: Alexandre Ern and Martin Vohralík
Journal: Math. Comp. 89 (2020), 551-594
MSC (2010): Primary 65N15, 65N30, 76M10
Published electronically: October 4, 2019
MathSciNet review: 4044442
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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 $ {\bm 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 $ {\bm 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.

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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

Martin Vohralík
Affiliation: Inria, 2 rue Simone Iff, 75589 Paris, France; and Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée 2, France

Keywords: Polynomial extension operator, broken Sobolev space, potential reconstruction, flux reconstruction, a posteriori error estimate, robustness, polynomial degree, best approximation, patch of elements
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.
Article copyright: © Copyright 2019 American Mathematical Society