L∞-Bounds for the L2-Projection onto Linear Spline Spaces

Abstract

In the univariate case, the L2-orthogonal projection PV 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 {double pipe}PV{double pipe} L∞→L∞ as established by Ciesielski, Oskolkov, and the author. As was shown more recently, the L2-orthogonal projection PV onto spaces V = V(T) of linear splines over triangulations 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 {double pipe}PV{double pipe} L∞→L∞ 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 PV in L∞ cannot be established. © Springer Science+Business Media, LLC 2013.