A Total Variation Diminishing Interpolation Operator and Applications
  We construct, on continuous $Q_1$ finite elements over Cartesian meshes, an interpolation operator that does not increase the total variation. The operator is stable in $L^1$ and exhibits second order approximation properties. With the help of it we provide improved error estimates for discrete minimizers of the total variation denoising problem and for total variation flows. 
Ricardo H. Nochetto and Abner J. Salgado
math.NA