A sharp Korn's inequality for piecewise $H^1$ space and its application

TL;DR

This paper establishes a sharp version of Korn's inequality for piecewise $H^1$ spaces with minimal jump conditions, aiding finite element analysis and construction.

Contribution

It introduces a minimal jump condition characterization for Korn's inequality on polygonal/polyhedral decompositions, demonstrating its sharpness and practical utility.

Findings

Minimal jump conditions are necessary and sufficient for Korn's inequality.

The conditions are explicitly characterized by rigid body modes on edges/faces.

The results facilitate testing and designing finite element spaces for Korn's inequality.

Abstract

In this paper, we revisit Korn's inequality for the piecewise $H^1$ space based on general polygonal or polyhedral decompositions of the domain. Our Korn's inequality is expressed with minimal jump terms. These minimal jump terms are identified by characterizing the restriction of rigid body mode to edge/face of the partitions. Such minimal jump conditions are shown to be sharp for achieving the Korn's inequality as well. The sharpness of our result 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.