## Polynomial extension operators for H1, Hcurl and Hdiv - spaces on a cube

*
Martin Costabel,
Monique Dauge,
Leszek Demkowicz*

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 H1,
- The tangential trace from Hcurl
- The normal trace from Hdiv.

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 Nédélec's 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.

Preprint IRMAR ** 07-15**.

Mathematics of Computation
**77** (2008), 1967-1999.

May 2007.

Slides Workshop ** Hofem ** (17-19 May 2007, Herrsching, Germany).