A counterexample concerning the $L_2$-projector onto linear spline spaces
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- by Peter Oswald PDF
- Math. Comp. 77 (2008), 221-226 Request permission
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 $\|P_V\|_{L_\infty \to L_\infty } \le 3$ 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.References
- I. Babuška and A. K. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal. 13 (1976), no. 2, 214–226. MR 455462, DOI 10.1137/0713021
- C. de Boor, On a max-norm bound for the least-squares spline approximant, Approximation and function spaces (Gdańsk, 1979) North-Holland, Amsterdam-New York, 1981, pp. 163–175. MR 649424
- Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141–157. MR 157182, DOI 10.4064/sm-23-2-141-157
- Z. Ciesielski, Private communication, Int. Conf. Approximation Theory and Probability, Bedlowo, 2004.
- Jean Descloux, On finite element matrices, SIAM J. Numer. Anal. 9 (1972), 260–265. MR 309292, DOI 10.1137/0709025
- Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in $L^{q}$ of the $L^{2}$-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193–197. MR 383789, DOI 10.1007/BF01400302
- A. Yu. Shadrin, The $L_\infty$-norm of the $L_2$-spline projector is bounded independently of the knot sequence: a proof of de Boor’s conjecture, Acta Math. 187 (2001), no. 1, 59–137. MR 1864631, DOI 10.1007/BF02392832
Additional Information
- Peter Oswald
- Affiliation: School of Engineering and Science, Jacobs University, D-28759 Bremen, Germany
- Email: poswald@jacobs-university.de
- Received by editor(s): December 20, 2006
- Published electronically: September 13, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 221-226
- MSC (2000): Primary 65N30, 41A15
- DOI: https://doi.org/10.1090/S0025-5718-07-02059-5
- MathSciNet review: 2353950