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A Counterexample concerning the L₂-Projector onto Linear Spline Spaces

Peter Oswald
Mathematics of Computation
Vol. 77, No. 261 (Jan., 2008), pp. 221-226
Stable URL: http://www.jstor.org/stable/40234504
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Counterexample concerning the L₂-Projector onto Linear Spline Spaces
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Abstract

For the L₂-orthogonal projection $P_v $ onto spaces of linear splines over simplicial partitions in polyhedral domains in ${\Cal R}^d ,d{\text{ \& }}gt;1$ , we show that in contrast to the one-dimensional case, where $\left\| {P_v } \right\|_{L_\infty \to L_\infty \,} \, \le \,3$ 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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