Numerical Results and Convergence of Some Inf-Sup Stable Elements for the Stokes Problem with Pressure Dirichlet Boundary Condition

For the Stokes problem with pressure Dirichlet boundary conditions, we propose an Enriched Mini element. For both the Mini element and the Enriched Mini element, we show that they are inf-sup stable. Unexpectedly, they yield wrong convergent finite element solutions for the singular velocity solution. On the contrary, the Taylor-Hood element, which is still inf-sup stable, gives correct convergence. However, how to analyze the convergence becomes open. We provide extensive numerical studies on the wrong convergence of the inf-sup stable Mini-type elements and the correct convergence of the inf-sup stable Taylor-Hood element and on the inf-sup stability constants.