Finite Element Approximation of Hydrostatic Stokes Equations: Review and Tests

Abstract

We present a review of a theory of stability and accuracy of Finite Element (FE) schemes for the Hydrostatic Stokes system which has been recently developed in Guillén-González and Rodríguez-Galván (Numer Math 130(2):225–256, 2015; SIAM J Numer Anal 53(4):1876–1896, 2015). Moreover, some new numerical results, not previously published, will be shown. This theory makes possible numerical simulations for classical FE (without the need of vertical integration required by most hydrostatic schemes in literature) and works even for anisotropic (not purely hydrostatic) models. The key is that stability of mixed approximation for Hydrostatic Stokes equations requires, besides the well-known Ladyzenskaja-Babuška-Brezzi (LBB) condition, an extra inf-sup condition. Some new numerical experiments are presented in this work. They suggest that for $(\mathcal{P}_{1}\! + \text{bubble})$–$\mathcal{P}_{1}$ one can reduce the number of degrees of freedom and also computational effort, without significantly worsening error orders. Some other unpublished numerical experiments are also presented here, in singular 2D domains and in realistic 3D domains (Gibraltar Strait).