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 H1-, L2- and L_infinity-norm for small data and convergence in H1-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; NewtonN"s method