On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra (extended version)

Abstract

For the reference triangle or tetrahedron $T$, we study the stability properties of the $L^2(T)$-projection ${\Pi }_N$ onto the space of polynomials of degree $N$. We show $\Vert {\Pi }_{N}u{\Vert }_{L^2(\partial T)}^2\le C\Vert u{\Vert }_{L^2(T)}\Vert u{\Vert }_{H^1(T)}$. This implies optimal convergence rates for the approximation error $\Vert u-{\Pi }_{N}u{\Vert }_{L^2(\partial T)}$ for all $u\in H^k(T)$, $k>1/2$.

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