Is $2k$-Conjecture valid for finite volume methods?

This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the bi-$k$ degree finite volume solution approximates the exact solution with an order $O(h^{2k})$, where $h$ is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments.