On the stability of the polynomial L-projection on triangles and tetrahedra

For the reference triangle or tetrahedron T , we study the stability properties of the L2(T )-projection ΠN onto the space of polynomials of degree N . We show ‖ΠNu‖L2(∂T ) ≤ C‖u‖L2(T )‖u‖H1(T ) and ‖ΠNu‖H1(T ) ≤ C(N + 1)‖u‖H1(T ). This implies optimal convergence rates for the approximation error ‖u−ΠNuu‖L2(∂T ) for all u ∈ Hk(T ), k > 1/2.