L _∞ -Bounds for the L ₂-Projection onto Linear Spline Spaces

Abstract

In the univariate case, the L ₂-orthogonal projection P _V onto a spline space V of degree k is bounded as an operator in L _∞ by a constant C(k) depending on the degree k but independent of the knot sequence. In the case of linear spline spaces the sharp bound is

$$\vert \vert {P}_{V }\vert {\vert }_{{L}_{\infty }\rightarrow {L}_{\infty }} < 3,$$

as established by Ciesielski, Oskolkov, and the author. As was shown more recently, the L ₂-orthogonal projection P _V onto spaces $V = V (\mathcal{T} )$ of linear splines over triangulations $\mathcal{T}$ of a bounded polygonal domain in ℝ² cannot be bounded in L _∞ by a constant that is independent of the underlying triangulation. Similar counterexamples show this for higher dimensions as well. In this note we state a new geometric condition on families of triangulations under which uniform boundedness of $\|{P{}_{V }\|}_{{L}_{\infty }\rightarrow {L}_{\infty }}$ can be guaranteed. It covers certain families of triangular meshes of practical interest, such as Shishkin and Bakhvalov meshes. On the other hand, we show that even for type-I triangulations of a rectangular domain uniform boundedness of P _V in L _∞ cannot be established.

Mathematics Subject Classification (2000): 65N30, 41A15.