  The Dirichlet-Neumann (DN) method has been extensively studied for linear
partial differential equations, while little attention has been devoted to the
nonlinear case. In this paper, we analyze the DN method both as a nonlinear
iterative method and as a preconditioner for Newton's method. We discuss the
nilpotent property and prove that under special conditions, there exists a
relaxation parameter such that the DN method converges quadratically. We
further prove that the convergence of Newton's method preconditioned by the DN
method is independent of the relaxation parameter. Our numerical experiments
further illustrate the mesh independent convergence of the DN method and
compare it with other standard nonlinear preconditioners.
