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 𝓟 2 /𝓟 1 or bubble 𝓟 1 b /𝓟 1 . Here we prove stability for our 𝓟 k /𝓟 k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.

中文翻译：

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静压Stokes问题稳定不连续Galerkin格式的数值分析

我们为静水 Stokes 方程提出了一种不连续伽辽金 (DG) 方案。这些方程与海洋学中的大尺度偏微分方程模型有关，其特点是垂直动量方程的椭圆度丢失。这一事实提供了一些有趣的挑战，例如稳定数值方案的设计。这里提出的新方案基于对称内罚 (SIP) 技术，对垂直速度进行了特殊处理。众所周知，原始方程的混合公式的稳定性除了 LBB inf-sup 条件外，还需要一个额外的与压力和垂直速度相关的流体静力 inf-sup 限制。这种流体静力 inf-sup 条件使通常的斯托克斯稳定连续有限元（如 Taylor-Hood 𝓟 2 /𝓟 1 或气泡 𝓟 1 b /𝓟 1 ）的稳定性无效。在这里，我们证明了我们的 𝓟 k /𝓟 k DG 方案的稳定性。提供了一些与前面的分析一致的新颖的数值试验。