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

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