In this section, we only consider one iteration of the back and forth nudging algorithm, i.e. one forward and one backward resolution with nudging terms. The idea consists of studying the decrease of the error (between the BFN trajectory and the true solution) during one iteration. All the results can then be extended very easily to an arbitrary number of BFN iterations. For instance, if the error decreases by a constant factor of less than 
during one iteration, then the algorithm is contractive and the error decreases exponentially to 0
with the iterations.
We refer to [29] for the proofs of all the following results.