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In this paper, the author derives an $O(h^4)$-superconvergence for thepiecewise linear Ritz-Galerkin finite element approximations for the secondorder elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichletboundary conditions. This superconvergence error estimate is establishedbetween the finite element solution and the usual Lagrange nodal pointinterpolation of the exact solution, and thus the superconvergence at the nodalpoints of each element. The result is based on a condition for the finiteelement partition characterized by the coefficient tensor $A$ and the usualshape functions on each element, called $A$-equilateral assumption in thispaper. Several examples are presented for the coefficient tensor $A$ and finiteelement triangulations which satisfy the conditions necessary forsuperconvergence. Some numerical experiments are conducted to confirm this newtheory of superconvergence.

Superconvergence of Ritz-Galerkin Finite Element Approximations for Second Order Elliptic Problems