On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra (extended version)
Provides a theoretical stability result for polynomial projections on simplices, relevant for finite element analysis and numerical methods.
The paper proves a stability bound for the L^2-projection onto polynomials on triangles and tetrahedra, showing that the boundary trace norm is bounded by the product of L^2 and H^1 norms, which yields optimal convergence rates for the approximation error on the boundary.
For the reference triangle or tetrahedron $T$, we study the stability properties of the $L^2(T)$-projection $Π_N$ onto the space of polynomials of degree $N$. We show $\|Π_N u\|_{L^2(\partial T)}^2 \leq C \|u\|_{L^2(T)} \|u\|_{H^1(T)}$. This implies optimal convergence rates for the approximation error $\|u - Π_N u\|_{L^2(\partial T)}$ for all $u \in H^k(T)$, $k > 1/2$.