Inf-sup stabilized Scott-Vogelius pairs on general simplicial grids for Navier-Stokes equations

Abstract: This paper considers the discretization of the time-dependent Navier-Stokes equations with the family of inf-sup stabilized Scott-Vogelius pairs recently introduced in [John/Li/Merdon/Rui, arXiv:2206.01242, 2022] for the Stokes problem. Therein, the velocity space is obtained by enriching the H^1-conforming Lagrange element space with some H(div)-conforming Raviart-Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a $P_k-P_0$-like system for arbitrary velocity polynomial degree $k$. Numerical studies verify the theoretical findings.