Volume 14, Issue 1
On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 31-46.

Published online: 2020-10

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• Abstract

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 $BV^2$($Ω$) which is useful for texture analysis and management in image restoration.

• Keywords

Poincaré-Friedrichs inequalities, broken Sobolev spaces, ${\rm C}^0$ Discontinuous Galerkin approximation, image processing.

65K10, 65N30, 68U10

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@Article{NMTMA-14-31, author = {R. H. W. Hoppe , }, title = {On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {31--46}, abstract = {

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 $BV^2$($Ω$) which is useful for texture analysis and management in image restoration.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0065}, url = {http://global-sci.org/intro/article_detail/nmtma/18326.html} }
TY - JOUR T1 - On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$ AU - R. H. W. Hoppe , 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 -

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 $BV^2$($Ω$) which is useful for texture analysis and management in image restoration.

R. H. W. Hoppe. (2020). On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 31-46. doi:10.4208/nmtma.OA-2020-0065
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