A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces 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 straightforward extension to barotropic situations with nonlinear equations of state is feasible. (C) 2020 Elsevier B.V. All rights reserved.