Generalized Korn's Inequalities for Piecewise $H^2$ Vector Fields

TL;DR

This paper introduces a new class of discrete generalized Korn's inequalities involving trace-free symmetric gradients for piecewise H2 vector fields, which could enhance finite element method analysis.

Contribution

It develops novel Korn's inequalities using trace-free symmetric gradients, extending the standard inequalities for better finite element method applications.

Findings

New inequalities involving trace-free symmetric gradients

Applicable to mixed finite element and discontinuous Galerkin methods

Potentially improves stability and error analysis in finite element methods

Abstract

The purpose of this paper is to construct a new class of discrete generalized Korn's inequalities for piecewise H2 vector fields in three-dimensional space. The resulting Korn's inequalities are different from the standard Korn's inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. It is anticipated that the new generalized Korn's inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.