Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

Abstract

SUMMARYThis paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier-Stokes equations. The method computes high-accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution (u, p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two-dimensional problems possessing a closed-form solution: a Poiseuille flow and a flow mimicking a boundary layer. 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. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique.