  A novel notion for constructing a well-balanced scheme - a gradient-robust
scheme - is introduced and a showcase application for a steady compressible,
isothermal Stokes equations is presented. Gradient-robustness means that
arbitrary gradient fields in the momentum balance are well-balanced by the
discrete pressure gradient - if there is enough mass in the system to
compensate the force. The scheme is asymptotic-preserving in the sense that it
degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust
discretization for the incompressible Stokes equations. The convergence of the
coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved
by compactness arguments. Numerical examples illustrate the numerical analysis,
and show that the novel approach can lead to a dramatically increased accuracy
in nearly-hydrostatic low Mach number flows. Numerical examples also suggest
that a straight-forward extension to barotropic situations with nonlinear
equations of state is feasible.
