Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes Problem

The supercloseness and superconvergence property of stabilized finite element methods applied to the Stokes problem are studied. We consider consistent residual based stabilization methods as well as nonconsistent local projection type stabilizations. Moreover, we are able to show the supercloseness of the linear part of the MINI-element solution which has been previously observed in practical computations. The results on supercloseness hold on three-directional triangular, axiparallel rectangular, and brick type meshes, respectively, but extensions to more general meshes are also discussed. Applying an appropriate postprocess to the computed solution, we establish superconvergence results. Numerical examples illustrate the theoretical predictions.