The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart--Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L2 and TV-L1, can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.
@article{arxiv.1804.07477,
title = {Discrete Total Variation with Finite Elements and Applications to Imaging},
author = {Marc Herrmann and Roland Herzog and Stephan Schmidt and José Vidal-Núñez and Gerd Wachsmuth},
journal= {arXiv preprint arXiv:1804.07477},
year = {2018}
}