The mixed finite element scheme of the Stokes problem with pressure stabilization is analyzed for the cross-grid Pk−Pk−1 elements, k≥1 , using discontinuous pressures. The Pk+−Pk−1 elements are also analyzed. We prove the stability of the scheme using the macroelement technique. The order of convergence follows from the standard theory of mixed methods. The macroelement technique can also be applicable to the stability analysis for some higher order methods using continuous pressures such as Taylor-Hood methods, cross-grid methods, or iso-grid methods.
Kim, Y., & Lee, S. (2000). Stable finite element methods for the Stokes problem. International Journal of Mathematics and Mathematical Sciences, 24(10), 699–714. https://doi.org/10.1155/s0161171200002908