  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.
