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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