Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations

We propose and justify the use of a simple scheme which recovers gradients from the piecewise linear finite-element approximation on triangular elements to the solution of a second-order elliptic problem. The recovered gradient is a superconvergent estimate of the true gradient at the midpoints of element edges. A related scheme recovers the gradient at the element centroids.