Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem

Abstract We propose a Discontinuous Galerkin (DG) scheme for the hydrostatic Stokes equations. These equations, related to large-scale PDE models in oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the symmetric interior penalty (SIP) technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of primitive equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝓟2/𝓟1 or bubble 𝓟1b/𝓟1. Here we prove stability for our 𝓟k/𝓟k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.