Superconvergent recovery operators: Derivative recovery techniques

To this date many gradient recovery techniques, which involve some post-processing of the gradient of the piecewise polynomial finite element solution for linear elliptic problems on planar domains, have been published. In many cases the effect of such a post-processing is to yield a pair of piecewise polynomial, possibly continuous, functions which approximate the gradient of the weak solution to an O(h) accuracy higher than the gradient of the finite element solution. In particular, for the case of piecewise linear finite element approximation, such a post-processing will result in a vector del(R)u(h), piecewise linear in both components, with the O(h(2)) accuracy compared to just O(h) accuracy of the usual gradient of the finite element solution under sufficient regularity of the weak solution of the given boundary value problem. In the present paper we study the approximation properties of the local derivatives of the recovered gradient functions. It is shown that these derivatives are O(h) approximations to the corresponding second derivatives of the weak solution when sufficient regularity of the weak solution is assumed. We also show that the Midpoint Recovered Gradient del(M)u(h), is equal to the gradient of some underlying quadratic u(h)(M) over each element; thereby allowing us to compute the second order h derivatives of u(h)(M) locally over each element. These second order derivatives are O(h) approximations to the corresponding second order derivatives of the weak solution in a discrete norm. We next repeat the recovery technique and obtain superconvergent second order derivatives under sufficient regularity assumptions. These recovered second order derivatives are differentiated once again to yield approximations to the third order derivatives of the weak solution over each element. We end this paper by supporting our study by means of a numerical example.