On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

Abstract

We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space W 2,1 (Ω; T h) with respect to a geometrically conforming, simplicial triagulation T h of a bounded Lipschitz domain Ω in R d , d ∈ N. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as C 0 Discontinuous Galerkin (C 0 DG) approximations of minimization problems in the Sobolev space W 2,1 (Ω), or more generally, in the Banach space BV 2 (Ω) of functions of bounded second order total variation. As an application, we consider a C 0 DG approximation of a minimization problem in BV 2 (Ω) which is useful for texture analysis and management in image restoration.