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-L 2 and TV-L 1 , 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.1. Introduction. The total-variation (TV)-seminorm | · | T V is ubiquitous as a regularizing functional in image analysis and related applications; see for instance [50,28,17,14]. When Ω ⊂ R 2 is a bounded domain, this seminorm is defined aswhere s ∈ [1, ∞], s * = s s−1 denotes the conjugate of s and | · | s * is the usual s * -norm of vectors in R 2 . Frequent choices include s = 2 (the isotropic case) and s = 1, see Figure 1.1.It has been observed in [21] that "the rigorous definition of the TV for discrete images has received little attention." In this paper we propose and analyze a discrete analogue of (1.1) for functions u belonging to a space DG r (Ω) or CG r (Ω) of globally discontinuous or continuous finite element functions of polynomial degree 1 0 ≤ r ≤ 4 on a geometrically conforming, simplicial triangulation of Ω, consisting of triangles T and interior edges E. 2 * This work was supported by DFG grants HE 6077/10-1 and SCHM 3248/2-1 within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged.