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

## TL;DR

This paper develops stable polynomial extension techniques in broken $H^1$ and ${f H}( ext{div})$ spaces for 3D problems, ensuring robustness with respect to polynomial degree, which aids in error analysis and flux reconstruction.

## Contribution

It introduces polynomial extensions in broken $H^1$ and ${f H}( ext{div})$ spaces with degree-independent stability constants, enhancing potential and flux reconstruction methods.

## Key findings

- Stability of polynomial extensions with degree-independent constants.
- Constructive proofs for extension stability.
- Application to polynomial-degree-robust a posteriori error analysis.

## 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 ${\bf H}(\mathrm{div})$ 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 ${\bf H}(\mathrm{div})$ 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.