We investigate Dirichlet–Neumann and Robin methods for a quasilinear elliptic transmission problem in which the nonlinearity changes discontinuously across two subdomains. In one space dimension, we obtain convergence theorems by extending known results from the linear case. They hold both on the continuous and on the discrete level. From the proofs one can infer mesh-independence of the convergence rates for the Dirichlet–Neumann method, but not for the Robin method. In two space dimensions, we consider numerical examples which demonstrate that the theoretical results might be extended to higher dimensions. Moreover, we investigate the asymptotic convergence behaviour for fine mesh sizes quantitatively. We observe a good agreement with many known linear results, which is remarkable in view of the nonlinear character of the problem.