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 Q₂ - P₁^disc element applied to the 3D stationary Stokes and Navier-Stokes problem, respectively. Moreover, applying a Q₃ - P₂^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 super-convergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost-benefit analysis between the two third-order methods, the post-processed Q₂ - P₁^disc discretization, and the Q ₃ - P₂^disc discretization is carried out.