TY - JOUR
T1 - On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$
AU - H. W. Hoppe , R.
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 31
EP - 46
PY - 2020
DA - 2020/10
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2020-0065
UR - https://global-sci.org/intro/article_detail/nmtma/18326.html
KW - Poincaré-Friedrichs inequalities, broken Sobolev spaces, ${\rm C}^0$ Discontinuous Galerkin
approximation, image processing.
AB - <p style="text-align: justify;">We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space $W^{2,1}$($Ω$; $\mathcal{T}_h$) with respect to a geometrically conforming, simplicial triagulation $\mathcal{T}_h$ of a bounded Lipschitz domain $Ω$ in $\mathbb{R}^d$ , $d$ $∈$ $\mathbb{N}$.
Such inequalities are of interest in the numerical analysis of nonconforming finite
element discretizations such as ${\rm C}^0$ Discontinuous Galerkin (${\rm C}^0$${\rm DG}$) approximations
of minimization problems in the Sobolev space $W^{2,1}$($Ω$), or more generally, in the
Banach space $BV^2$($Ω$) of functions of bounded second order total variation. As
an application, we consider a ${\rm C}^0$${\rm DG}$ approximation of a minimization problem in&nbsp;$BV^2$($Ω$) which is useful for texture analysis and management in image restoration.</p>