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

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.