A posteriori error estimators for regularized total variation of characteristic functions

We consider a non-uniformly elliptic double obstacle problem arising from ``convexifying'' and regularizing a minimization of a functional with total variation in the set of characteristic functions. We derive a posteriori estimators for the discretization error with linear finite elements, which are uniform in the regularization, incorporate computable and local information on the conditioning, vanish in the intersection of discrete and exact contact set, and are not affected by possible non-uniqueness. Moreover, we integrate these estimators in an adaptive algorithm and illustrate their properties by various numerical experiments.