## A Finite Volume Element Method for Nonlinear Elliptic Problems

**Abstract.**
We consider a finite volume discretization of second-order non-linear elliptic
boundary value problems on polygonal domains. Using relatively standard assumptions
we show the existence of the finite volume solution. Furthermore, for a sufficiently
small data the uniqueness of the finite volume solution may also be deduced. We
derive error estimates in $H^1$-, $L_2$- and $L_\infty$-norm for small data and
convergence in $H^1$-norm for large data. In
addition a Newton's method is analysed for the approximation of the finite
volume solution and numerical experiments are presented.

**Keywords.**
finite volume element method; non-linear elliptic equation; error estimates; fixed point
iterations; Newton's method