  In this article we investigate a finite element formulation of strongly
monotone quasi-linear elliptic PDEs in the context of fixed-point iterations.
As opposed to Newton's method, which requires information from the previous
iteration in order to linearise the iteration matrix (and thereby to recompute
it) in each step, the alternative method used in this article exploits the
monotonicity properties of the problem, and only needs the iteration matrix
calculated once for all iterations of the fixed-point method. We outline the a
priori and a posteriori error estimates for iteratively obtained solutions, and
show both theoretically as well as numerically how the number of iterations of
the fixed-point method can be restricted in dependence of the mesh size, or of
the polynomial degree, to obtain optimal convergence.
