| In this talk, we shall report a new a posteriori error estimate based on superconvergence estimates for piecewise linear finite element approximations on nonuniform but smoothly varying triangular meshes. We first show the finite element solution $u_h$ and the interpolant $u_I$ have super close gradients for a special class of grids. We then develop a postprocessing gradient recovery scheme for $u_h$, inspired in part by the smoothing iteration of the multigrid method. We show that this recovered gradient superconverges to that of the true solution for general quasi-uniform grids. Next, we use the supconvergent gradient to approximate the Hessian matrix of the true solution, and form an a posteriori error estimate that is is in some sense asymptotically exact. |