  We propose a globally convergent numerical method to compute solutions to a
general class of quasi-linear PDEs with both Neumann and Dirichlet boundary
conditions. Combining the quasi-reversibility method and a suitable Carleman
weight function, we define a map of which fixed point is the solution to the
PDE under consideration. To find this fixed point, we define a recursive
sequence with an arbitrary initial term using the same manner as in the proof
of the contraction principle. Applying a Carleman estimate, we show that the
sequence above converges to the desired solution. On the other hand, we also
show that our method delivers reliable solutions even when the given data are
noisy. Numerical examples are presented.
