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:
  1. The standard one from H1,
  2. The tangential trace from Hcurl
  3. 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.

Pdf file (336 k)

May 2007.

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

Pdf file (144 k)