Gradient-robustness for the isentropic compressible Stokes problem

This paper extends a scheme for the isothermal compressible Stokes problem to the isentropic compressible Stokes problem that has two important ingredients. The first one is a reconstruction operator in the right-hand side that improves the correct balancing of gradient forces in hydrostatic situations. This so-called gradient-robustness makes the scheme asymptotics-preserving (AP) in the zero Mach number limit in the sense that it converges on fixed grids to a pressure-robust scheme for the incompressible Stokes problem. The second ingredient is an AP-compatible stabilization term in the continuity equation that allows a-priori provable strong convergence of the scheme, in particular the satisfaction of the nonlinear barotropic equation of state in the limit. The paper explains the algorithmical details, proves stability and convergence of the scheme as well as meaningful error estimates for the velocity. The results are illustrated by some numerical examples.