In recent years, nonlinear domain decomposition methods (DDMs) have become more and more popular. They are characterized by a decomposition of the nonlinear problem before linearization - usually with Newton's method. If set up appropriately, nonlinear DDMs show a faster nonlinear convergence and a larger convergence radius than Newton-Krylov methods equipped with a linear DD or multigrid preconditioner (Newton-Krylov-DDMs). Additionally, nonlinear DDMs tend to increase the local work while reducing the amount of communication in parallel implementations. Therefore, they have the potential to reduce the computing time and improve parallel scalability. Nevertheless, they cannot use modern hardware (as GPUs) more efficiently than classical linear DDMs since they still rely on many computations with large and distributed sparse matrices. In this talk, specifically nonlinear two-level Schwarz methods with GDSW (Generalized Dryja-Smith-Widlund) coarse spaces are considered. It is discussed how these methods can be used to reduce the amount of global sparse matrix computations. To reach this goal, alternatives to Newton's method are exploited for all global computations and it is showed that similar ideas cannot be used in a classical Newton-Krylov-DDM framework. In other words, using nonlinear DDMs can be beneficial in the context of matrix-free nonlinear preconditioners. Finally, new coarse spaces based on new nonlinear extensions, which further improve the nonlinear convergence, are introduced.