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Mathematics of Computation

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A counterexample concerning the $ L_2$-projector onto linear spline spaces


Author: Peter Oswald
Journal: Math. Comp. 77 (2008), 221-226
MSC (2000): Primary 65N30, 41A15
DOI: https://doi.org/10.1090/S0025-5718-07-02059-5
Published electronically: September 13, 2007
MathSciNet review: 2353950
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Abstract: For the $ L_2$-orthogonal projection $ P_V$ onto spaces of linear splines over simplicial partitions in polyhedral domains in $ \mathbb{R}^d$, $ d>1$, we show that in contrast to the one-dimensional case, where $ \Vert P_V\Vert _{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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Additional Information

Peter Oswald
Affiliation: School of Engineering and Science, Jacobs University, D-28759 Bremen, Germany
Email: poswald@jacobs-university.de

DOI: https://doi.org/10.1090/S0025-5718-07-02059-5
Received by editor(s): December 20, 2006
Published electronically: September 13, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.