Construction of polynomial extensions in two dimensions and application to the $h\text{-}p$ finite element method

Abstract

Polynomial extensions play a vital role in the analysis of the $p$ and $h\text{-}p$ FEM as well as the 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 $\Gamma$ or on whole boundary of $T$ or $S$. The continuity of these extension operators from $H_{00}^{\frac{1}{2}}\left(\Gamma \right)$ to $H^1\left(T\right)$ or $H^1\left(S\right)$ and from $H^{\frac{1}{2}}\left(\partial T\right)$ to $H^1\left(T\right)$ or from $H^{\frac{1}{2}}\left(\partial S\right)$ to $H^1\left(S\right)$ is rigorously proved in a constructive way. Applications of these polynomial extensions to the error analysis for the $h\text{-}p$ FEM are presented.