We investigate optimization problems with a non-smooth partial differential equation as constraint, where the non-smoothness is assumed to be caused by Nemytzkii operators generated by the functions abs, min and max. For the efficient as well as robust solution of such problems, we propose a new optimization method based on abs-linearization, i.e., a special handling of the non-smoothness with proficient exploitation of the non-smooth structure. The exploitation of the given data allows a targeted and optimal decomposition of the optimization problem in order to compute stationary points. This approach is able to solve the considered class of non-smooth optimization problems in comparably less Newton steps and additionally maintains reasonable convergence properties. Numerical results for non-smooth optimization problems illustrate the proposed approach and its performance.