Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem
In this paper, the stability of a discrete interior penalty discontinuous Galerkin (DG) is investigated for the 3D steady hydrostatic Stokes equations, which describe geophysical flows of the ocean and the atmosphere. Due to the loss of ellipticity of the vertical momentum equation, standard finite element approximations are not well suited. The new scheme is based on the standard symmetric interior penalty (SIP) technique for the horizontal momentum equations, with a particular penalization for the vertical component of velocity, where only the jumps in the vertical direction is considered. It is known that stability of the mixed formulation of primitive equations requires, besides the well-known LBB (Ladyzhenskaya-Babuška-Brezzi) inf-sup condition, an additional hydrostatic inf-sup restriction relating the pressure and the vertical velocity. The authors show by using the same polynomial order for velocity and pressure, discrete versions of both constraints, the LBB and the hydrostatic inf-sup restrictions, are satisfied, which provides well-posedness and stability of the scheme. Numerical tests illustrate the applicability of scheme to practical cases, such as for canonical cavity problems
scientific article; zbMATH DE number 7394126
scientific article