This paper deals with discrete analogues of nonlinear elliptic boundary value problems and with monotonically convergent iterative methods for their numerical solution. The discrete analogues can be written asM(u)u+H(u)=0, whereM(u) is ann%n M-matrix for eachu?? n andH: ? n ?? n . The numerical methods considered are the `natural undeerrelaxation method', the `successive underrelaxation method', and the `Jacobi underrelaxation method'. In the linear case and without underrelaxation these methods correspond to the direct, the Gauss-Seidel, and the Jacobi method for solving the underlying system of equations, resp. For suitable starting vectors and sufficiently strong underrelaxation, the sequence of iterates generated by any of these methods is shown to converge monotonically to a solution of the underlying system.