A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

Numerical Analysis of a Stable Discontinuous Galerkin Scheme for the Hydrostatic Stokes Problem

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 SIP penalty 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.