  In this paper, we analyze vertex-centered finite volume method (FVM) of any
order for elliptic equations on rectangular meshes. The novelty is a unified
proof of the inf-sup condition, based on which, we show that the FVM
approximation converges to the exact solution with the optimal rate in the
energy norm. Furthermore, we discuss superconvergence property of the FVM
solution. With the help of this superconvergence result, we find that the FVM
solution also converges to the exact solution with the optimal rate in the
$L^2$-norm. Finally, we validate our theory with several numerical experiments.
