The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

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: ${\max }_{P\in S}\mathrm{\mid }(\mathrm{\nabla }u-\bar{\mathrm{\nabla }}{u}_h)(P)\mathrm{\mid }=O(h^2)\mathrm{\mid }\ln h\mathrm{\mid }$, where $\bar{\mathrm{\nabla }}{u}_h(P)$ 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.