Polynomial extension operators for ๐ปยน, ๐ป(๐๐ฎ๐ซ๐ฅ) and ๐(๐๐ข๐ฏ)-spaces on a cube
Martin Costabel, Monique Dauge, L. Demkowicz ยท Mathematics of Computation ยท 2008
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 H^1 , the tangential trace from H ( c u r l ) \boldsymbol {H}(\mathbf {curl}) and the normal trace from H ( d i v ) \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 Nรฉdรฉlec hexahedron of the first type of degree p p . The core of the paper is the construction of polynomial trace liftings with operator norms bounded independently of the polynomial degree p 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.