Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations

Recently, it was understood how to repair a certain $L^2$-orthogonality of discretely divergence-free vector fields and gradient fields such that the velocity error of inf–sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new ‘pressure-robust’ Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf–sup stable Stokes discretizations can guarantee a small velocity error only when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, ‘pressure-robustness’ is extended to the time-dependent Navier–Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the ‘discrete Helmholtz projector’ of an inf–sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed