(untitled) (Journal of Computational Mathematics, 2026)
In this paper, two O(h^2)-accurate conservative finite element schemes with low-degree polynomials for the incompressible Stokes equations are presented. The schemes use respective H(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 sBDM_3−P_2 and sBDFM_3−P_2 schemes, respectively. The discrete Korn inequality holds for both sBDM_3 and sBDFM_3 finite element spaces. For the sBDM_3−P_2 scheme, the inf-sup condition holds on general triangulations, and for the sBDFM_3−P_2 scheme, the inf-sup condition holds on triangulations with mild restriction. Both schemes achieve an energy norm of velocity errors of O(h^2) order and an L^2-norm of pressure errors of O(h^2) order. Numerical experiments support the theoretical constructions.