Polynomial extension operators for 𝐻¹, 𝐻(𝐜𝐮𝐫𝐥) and 𝐇(𝐝𝐢𝐯)-spaces on a cube

Costabel, M.; Dauge, M.; Demkowicz, L.

doi:10.1090/S0025-5718-08-02108-X

Polynomial extension operators for $H^1$, $\boldsymbol {H}(\mathbf {curl})$ and $\boldsymbol {H}(\mathbf {div})$ - spaces on a cube
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by M. Costabel, M. Dauge and L. Demkowicz PDF

Math. Comp. 77 (2008), 1967-1999 Request permission

Abstract:

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 $H^1$, the tangential trace from $\boldsymbol {H}(\mathbf {curl})$ and the normal trace from $\boldsymbol {H}(\mathrm {div})$. 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 the Nédélec 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.

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