Equal-order Finite Elements for the Hydrostatic Stokes Problem

Abstract

Simulation of flow phenomena in the ocean and in other large but relatively flat basins are typically based on the so-called primitive equations, which, among others, result from application of the hydrostatic approximation. The crucial premise for this approximation is the dominance of the hydrostatic balance over remaining vertical flow phenomena in large but flat domains, which leads to a decomposition of the three-dimensional (3D) pressure field into a hydrostatic part and an only two-dimensional (2D) hydrodynamic part. The former pressure can be obtained by solving Ordinary Differential Equations. The latter one is determined by a 2D elliptic problem which can be solved quite efficiently. The velocity field remains three dimensional. However, its vertical component can be eliminated from the dynamic system. In this work, we analyze such "2.5-dimensional" (2.5D) Stokes systems and formulate stabilized finite element schemes with equal-order interpolation. The absence of a discrete inf-sup condition is compensated by introducing additional terms into the discrete variational form. We show stability and give an a priori error estimate for several established stabilized equal-order schemes, as pressure-stabilized Petrov-Galerkin (PSPG), Galerkin least squares (GLS) and local projection schemes (LPS) which are extended here to the hydrostatic approximation. The basic assumption we need is a certain property of the underlying 3D mesh. We illustrate the order of convergence of the 2.5D problem by numerical examples and demonstrate the effect of the hydrostatic approximation in comparison to the full 3D problem.