AbstractIn 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.
A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation
