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

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 25)


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.


Uniform Boundedness Polygonal Domain Spline Space Rectangular Cell Shishkin Mesh 
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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Jacobs University BremenBremenGermany

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