On The Sharpness of a Korn’s Inequality For Piecewise H1 Space and Its Applications

Qingguo Hong; Young Ju Lee; Jinchao Xu

doi:10.1007/s10915-024-02724-w

On The Sharpness of a Korn’s Inequality For Piecewise H1 Space and Its Applications

Qingguo Hong^*, Young Ju Lee, Jinchao Xu

^*Corresponding author for this work

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

Abstract

In this paper, we investigate the sharpness of a Korn’s inequality for piecewise H¹ space and its applications. We first revisit a Korn’s inequality for the piecewise H¹ space based on general polygonal or polyhedral decompositions of the domain. We express the Korn’s inequality with minimal jump terms. Then we prove that such minimal jump conditions are sharp for achieving the Korn’s inequality. The sharpness of the Korn’s inequality and explicitly given minimal conditions can be used to test whether any given finite element spaces satisfy Korn’s inequality, immediately as well as to build or modify nonconforming finite elements for Korn’s inequality to hold.

Original language	English (US)
Article number	6
Journal	Journal of Scientific Computing
Volume	102
Issue number	1
DOIs	https://doi.org/10.1007/s10915-024-02724-w
State	Published - Jan 2025

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

Keywords

Korn’s inequality
Piecewise space
Sharpness

ASJC Scopus subject areas

Software
Theoretical Computer Science
Numerical Analysis
General Engineering
Computational Mathematics
Computational Theory and Mathematics
Applied Mathematics

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10.1007/s10915-024-02724-w

Cite this

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title = "On The Sharpness of a Korn{\textquoteright}s Inequality For Piecewise H1 Space and Its Applications",

abstract = "In this paper, we investigate the sharpness of a Korn{\textquoteright}s inequality for piecewise H1 space and its applications. We first revisit a Korn{\textquoteright}s inequality for the piecewise H1 space based on general polygonal or polyhedral decompositions of the domain. We express the Korn{\textquoteright}s inequality with minimal jump terms. Then we prove that such minimal jump conditions are sharp for achieving the Korn{\textquoteright}s inequality. The sharpness of the Korn{\textquoteright}s inequality and explicitly given minimal conditions can be used to test whether any given finite element spaces satisfy Korn{\textquoteright}s inequality, immediately as well as to build or modify nonconforming finite elements for Korn{\textquoteright}s inequality to hold.",

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On The Sharpness of a Korn’s Inequality For Piecewise H1 Space and Its Applications

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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