Pressure-robustness by commuting interpolation operators for Stokes discretizations with continuous pressures

Abstract

Common finite element discretizations for the incompressible Stokes equations with a continuous pressure approximation -- the MINI element, the Taylor--Hood element, or stabilized equal-order elements -- are not pressure-robust: the velocity error is polluted by the pressure best-approximation error, multiplied by the inverse viscosity. The established remedy replaces the test function in the momentum equation by a divergence-preserving reconstruction operator; for continuous pressure approximations this construction is cumbersome, requiring the solution of local problems (Lederer, Linke, Merdon&Sch\"oberl, SIAM J. Numer. Anal., 2017). We propose a simpler alternative: interpolating the force by a N\'ed\'elec interpolation operator of suitable order instead of reconstructing the test function. Exploiting the commuting diagram property linking N\'ed\'elec and Lagrange interpolation operators, we show that the resulting method is pressure-robust for any $H^1$-conforming, continuous-pressure discretization. We give the complete analysis for the MINI element, and then for stabilized equal-order $P^kP^k$ elements of arbitrary degree $k$, proving that the interpolation-induced consistency error is pressure-robust and converges of optimal order. We validate the theory by several numerical examples.