  In this paper, the author derives an $O(h^4)$-superconvergence for the
piecewise linear Ritz-Galerkin finite element approximations for the second
order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet
boundary conditions. This superconvergence error estimate is established
between the finite element solution and the usual Lagrange nodal point
interpolation of the exact solution, and thus the superconvergence at the nodal
points of each element. The result is based on a condition for the finite
element partition characterized by the coefficient tensor $A$ and the usual
shape functions on each element, called $A$-equilateral assumption in this
paper. Several examples are presented for the coefficient tensor $A$ and finite
element triangulations which satisfy the conditions necessary for
superconvergence. Some numerical experiments are conducted to confirm this new
theory of superconvergence.
