  Recent analysis of the divergence constraint in the incompressible
Stokes/Navier--Stokes problem has stressed the importance of equivalence
classes of forces and how it plays a fundamental role for an accurate space
discretization. Two forces in the momentum balance are velocity--equivalent if
they lead to the same velocity solution, i.e., if and only if the forces differ
by only a gradient field. Pressure-robust space discretizations are designed to
respect these equivalence classes. One way to achieve pressure-robust schemes
is to introduce a non-standard discretization of the right-side forcing term
for any inf-sup stable mixed finite element method. This modification leads to
pressure-robust and optimal-order discretizations, but a proof was only
available for smooth situations and remained open in the case of minimal
regularity, where it cannot be assumed that the vector Laplacian of the
velocity is at least square-integrable. This contribution closes this gap by
delivering a general estimate for the consistency error that depends only on
the regularity of the data term. Pressure-robustness of the estimate is
achieved by the fact that the new estimate only depends on the $L^2$ norm of
the Helmholtz--Hodge projector of the data term and not on the $L^2$ norm of
the entire data term. Numerical examples illustrate the theory.

Numerical Analysis;Computational Physics