Generalized Korn’s inequalities for piecewise 𝐻¹ and 𝐻² vector fields

Abstract

The purpose of this paper is to construct a new class of discrete generalized Korn’s inequalities for piecewise H 1 H^1 vector fields and piecewise H 2 H^2 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.