Superconvergence of a 3D finite element method for stationary Stokes and Navier-Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q(2) - P-1(disc) element applied to the 3D stationary Stokes and Navier-Stokes problem, respectively. Moreover, applying a Q(3) - P-2(disc) postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q(2)-interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost-benefit analysis between the two third-order methods, the post-processed Q(2) - P-1(disc) discretization, and the Q(3) - P-2(disc) discretization is carried out. (c) 2004 Wiley Periodicals, Inc.