  We study the gradient superconvergence of bilinear finite volume element
(FVE) solving the elliptic problems. First, a superclose weak estimate is
established for the bilinear form of the FVE method. Then, we prove that the
gradient approximation of the FVE solution has the superconvergence property:
$\max_{P\in S}|(\nabla u-\overline{\nabla}u_h)(P)|=O(h^2)|\ln h|$, where
$\overline{\nabla}u_h(P)$ denotes the average gradient on elements containing
point $P$ and $S$ is the set of optimal stress points composed of the mesh
points, the midpoints of edges and elements.
