We study a narrow band type algorithm to solve a discrete formulation of the convex relaxation of energy functionals with total variation regularization and nonconvex data terms. We prove that this algorithm converges to a local minimum of the original nonlinear optimization problem. We illustrate the algorithm with experiments for disparity computation in stereo and a multilabel segmentation problem, and we check experimentally that the energy of the local minimum is very near to the energy of the global minimum obtained without the narrow band type method.