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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.