WIAS Preprint No. 2641, (2019)

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


  • Akbas, Mine
  • Gallouët, Thierry
  • Gaßmann, Almut
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 76D07 65N30 65N12

2010 Physics and Astronomy Classification Scheme

  • 47.10.ad 47.11.Fg


  • compressible Stokes equations, finite element method, finite volume method, well-balanced scheme, upwind, convergence




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.

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