Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

Èrgash Muhamadiev; Murtazo Nazarov

doi:10.1016/j.jmaa.2014.10.027

Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

Èrgash Muhamadiev, Murtazo Nazarov

Research output: Contribution to journal › Article › peer-review

Abstract

© 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p$^{-2}$, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln$^{2}$ p which is an increasing function. Moreover, we prove that this estimate is sharp.

Original language	English (US)
Pages (from-to)	940-955
Number of pages	16
Journal	Journal of Mathematical Analysis and Applications
Volume	423
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2014.10.027
State	Published - Mar 2015
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The second author is supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1016/j.jmaa.2014.10.027

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title = "Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions",

abstract = "{\textcopyright} 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p$^{-2}$, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln$^{2}$ p which is an increasing function. Moreover, we prove that this estimate is sharp.",

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doi = "10.1016/j.jmaa.2014.10.027",

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AU - Nazarov, Murtazo

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N2 - © 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p$^{-2}$, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln$^{2}$ p which is an increasing function. Moreover, we prove that this estimate is sharp.

AB - © 2014 Elsevier Inc. In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p$^{-2}$, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln$^{2}$ p which is an increasing function. Moreover, we prove that this estimate is sharp.

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

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