Construction of polynomial extensions in two dimensions and application to the h-ph-p finite element method

Polynomial extensions play a vital role in the analysis of the pp and h-ph-p FEM as well as the spectral element method. In this paper, we construct explicitly polynomial extensions on a triangle TT and a square SS, which lift a polynomial defined on a side ΓΓ or on whole boundary of TT or SS. The continuity of these extension operators from H0012(Γ) to H1(T)H1(T) or H1(S)H1(S) and from H12(∂T) to H1(T)H1(T) or from H12(∂S) to H1(S)H1(S) is rigorously proved in a constructive way. Applications of these polynomial extensions to the error analysis for the h-ph-p FEM are presented.