A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

We present a mixed finite element method for the steady-state Stokes equations where the discrete bilinear form for the velocity is obtained by a weakly over-penalized symmetric interior penalty approach. We show that this mixed finite element method is inf-sup stable and has optimal convergence rates in both the energy norm and the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} norm on meshes that can contain hanging nodes. We present numerical experiments illustrating these results, explore a very simple adaptive algorithm that uses meshes with hanging nodes, and introduce a simple but scalable parallel solver for the method.