  The Neumann--Neumann method is a commonly employed domain decomposition
method for linear elliptic equations. However, the method exhibits slow
convergence when applied to semilinear equations and does not seem to converge
at all for certain quasilinear equations. We therefore propose two modified
Neumann--Neumann methods that have better convergence properties and require
less computations. We provide numerical results that show the advantages of
these methods when applied to both semilinear and quasilinear equations. We
also prove linear convergence with mesh-independent error reduction under
certain assumptions on the equation. The analysis is carried out on general
Lipschitz domains and relies on the theory of nonlinear Steklov--Poincar\'e
operators.
