Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations

In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy.