Constructive Proof for Polynomial Extensions in Two Dimensions and Application to the h-p Finite Element Method

Polynomial extensions play a vital role in analysis of the p and h-p FEM as well as spectral element method. In this paper, we construct explicitly polynomial extensions on a triangle T and a square S, which lift a polynomial defined on a side Γ or on whole boundary of T or S. The continuity of these extension operators from H 1 2 00(Γ) to H (T ) or H(S) and from H 1 2 (∂T ) to H(T ) or from H 1 2 (∂S) to H(S) is rigorously and constructively proved. Applications of these polynomial extensions to the error analysis for the h-p FEM are presented.