Stable and Compatible Polynomial Extensions in Three Dimensions and Applications to the p and h-p Finite Element Method

Polynomial extensions play a vital role in the analysis of the p and h-p FEM and the spectral element method. We construct explicitly polynomial extensions on standard elements: cubes, triangular prisms and pyramids, which together with the extension on tetrahedrons are used by the p and h-p FEM in three dimensions. These extensions are proved to be stable and compatible with FEM subspaces on tetrahedrons, cubes, prisms and pyramids, and realize a continuous mapping: H 1/2 00 (T )( or H 1/2 00 (S)) → H(Ωst), where Ωst denotes one of these standard elements and T and S are their triangular and square faces. Applications of these polynomial extensions to the p and h-p FEM are illustrated.