A counterexample concerning the L2-projector onto linear spline spaces

For the L2-orthogonal projection PV onto spaces of linear splines over simplicial partitions in polyhedral domains in Rd, d > 1, we show that in contrast to the one-dimensional case, where ‖PV ‖L∞→L∞ ≤ 3 independently of the nature of the partition, in higher dimensions the L∞-norm of PV cannot be bounded uniformly with respect to the partition. This fact is folklore among specialists in finite element methods and approximation theory but seemingly has never been formally proved.