Computation of accurate nodal derivatives of finite element solutions: the finite node displacement method

This paper presents a new method for extracting high‐accuracy nodal derivatives from finite element solutions. The approach involves imposing a finite displacement to individual mesh nodes, and solving a very small problem on the patch of surrounding elements, whose only unknown is the value of the solution at the displaced node. A finite difference between the original and perturbed values provides the directional derivative. Verification is shown for a one‐dimensional diffusion problem with exact nodal solution and for two‐dimensional scalar advective–diffusive problems. For internal nodes the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. In this case, the local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. We show that the new method gives normal derivatives at boundary points that are consistent with the so‐called ‘ auxiliary fluxes ’. The resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique.