Superconvergent Derivative Recovery for Lagrange Triangular Elements of Degree p on Unstructured Grids
by Randolph E. Bank, Jinchao Xu, Bin Zheng
In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution $u_h$ on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree $p$, by applying the global $L^2$ projection ($Q_h$) and a smoothing operator ($S_h$), the recovered $p$-th derivative ($S_h^m Q_h\partial^p u_h$) superconverges to the exact derivatives ($\partial^p u$). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the $(p+1)$-st derivative approximated by $\partial S_h^m Q_h\partial^p u_h$. We provide several numerical examples illustrating the effectiveness of our procedures. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence.