Low-Degree $\mathcal{O}(h^2)$-Accurate Conservative Finite Element Schemes for Incompressible Stokes Problems on General Triangulations

Abstract

In this paper, two $\mathcal{O}(h^2)$-accurate conservative finite element schemes with low-degree polynomials for the incompressible Stokes equations are presented. The schemes use respective $H({\rm div})$ finite element spaces, namely the third-order Brezzi-Douglas-Marini space and Brezzi-Douglas-Fortin-Marini space, with enhanced smoothness for the velocity and piecewise quadratic polynomials for the pressure, and are denoted as ${\rm sBDM}_3−{\rm P}_2$ and ${\rm sBDFM}_3−{\rm P}_2$ schemes, respectively. The discrete Korn inequality holds for both ${\rm sBDM}_3$ and ${\rm sBDFM}_3$ finite element spaces. For the ${\rm sBDM}_3−{\rm P}_2$ scheme, the inf-sup condition holds on general triangulations, and for the ${\rm sBDFM}_3−{\rm P}_2$ scheme, the inf-sup condition holds on triangulations with mild restriction. Both schemes achieve an energy norm of velocity errors of $\mathcal{O}(h^2)$ order and an $L^2$-norm of pressure errors of $\mathcal{O}(h^2)$ order. Numerical experiments support the theoretical constructions.