A counterexample concerning the 𝐿₂-projector onto linear spline spaces

Oswald, Peter

doi:10.1090/S0025-5718-07-02059-5

A counterexample concerning the $L_2$-projector onto linear spline spaces
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Math. Comp. 77 (2008), 221-226 Request permission

Abstract:

For the $L_2$-orthogonal projection $P_V$ onto spaces of linear splines over simplicial partitions in polyhedral domains in $\mathbb {R}^d$, $d>1$, we show that in contrast to the one-dimensional case, where $\|P_V\|_{L_\infty \to L_\infty } \le 3$ independently of the nature of the partition, in higher dimensions the $L_\infty$-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.

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