## Finite Volume Element Method for Second-Order Quasilinear Elliptic Problems

**Abstract.**
In this paper, we consider the finite volume element method for
the general second-order quasilinear elliptic problems over convex
polygonal domain in the plane. Using reasonable assumptions, we show
the existence and uniqueness of the finite volume element
approximations. It is proved that the finite volume element
approximations are convergent with $\mathcal{O}(h)$,
$\mathcal{O}(h^{1-2/r} |\ln h|)$, for $r>2$, and
$\mathcal{O}(h^2|\ln h|)$ in the $H^1$-,
$W^{1,\infty}$-, and $L^2$-norm, respectively. Moreover, the standard
$\mathcal{O}(h^2)$ convergence rate in the $L^2$-norm is derived
for some special quasilinear elliptic problems. Numerical
experiments are presented to confirm the estimates.