A counterexample concerning the $L_2$-projector onto linear spline spaces

Abstract

For the L 2 -orthogonal projection P V onto spaces of linear splines over simplicial partitions in polyhedral domains in R a , d > 1, we show that in contrast to the one-dimensional case, where ∥P v ∥L∞→L∞ < 3 independently of the nature of the partition, in higher dimensions the L∞-norm of P V 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.