Solution of monotone nonlinear elliptic boundary value problems

LetL be a linear uniformly elliptic second order operator. The boundary value problem is solved for the nonlinear elliptic equationLu=f(x, u) wheref(x, u) is a monotone increasing function ofu for each pointx in the domain. A descent technique based on Newton's method is shown to yield a sequence of iterates which converges uniformly and quadratically to the solution. The convergence is independent of the choice for the initial iterate. Numerical results in two dimensions are presented.