We construct an interpolation operator that does not increase the total variation and is defined on continuous first degree finite elements over Cartesian meshes for any dimension d d and right triangular meshes for d = 2 d = 2 . The operator is stable and exhibits second order approximation properties in any L p L^p , 1 ≤ p ≤ ∞ 1\leq p \leq \infty . With the help of it we provide improved error estimates for discrete minimizers of the total variation denoising problem and for total variation flows. We also explore computationally the limitations of the total variation diminishing property over non-Cartesian meshes.