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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)

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.

Keywords

Uniform Boundedness Polygonal Domain Spline Space Rectangular Cell Shishkin Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Jacobs University BremenBremenGermany

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