Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem
type: Item
instance of: scholarly article
title: Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem
zbMATH Open document ID: 1480.65257
author: María Victoria Redondo-Neble
author: J. Rafael Rodríguez-Galvàn
author: Francisco Guillén-González
published in: Journal of Numerical Mathematics
publication date: 2021-09-10T00:00:00Z
review text: In this paper, the stability of a discrete interior penalty discontinuous Galerkin (DG) is investigated for the 3D steady hydrostatic Stokes equations, which describe geophysical flows of the ocean and the atmosphere. Due to the loss of ellipticity of the vertical momentum equation, standard finite element approximations are not well suited. The new scheme is based on the standard symmetric interior penalty (SIP) technique for the horizontal momentum equations, with a particular penalization for the vertical component of velocity, where only the jumps in the vertical direction is considered. It is known that stability of the mixed formulation of primitive equations requires, besides the well-known LBB (Ladyzhenskaya-Babuška-Brezzi) inf-sup condition, an additional hydrostatic inf-sup restriction relating the pressure and the vertical velocity. The authors show by using the same polynomial order for velocity and pressure, discrete versions of both constraints, the LBB and the hydrostatic inf-sup restrictions, are satisfied, which provides well-posedness and stability of the scheme. Numerical tests illustrate the applicability of scheme to practical cases, such as for canonical cavity problems
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 76U05
Mathematics Subject Classification ID: 86A05
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35Q86
zbMATH DE Number: 7394126
zbMATH Keywords: discontinuous Galerkin
zbMATH Keywords: hydrostatic Stokes
zbMATH Keywords: numerical analysis
zbMATH Keywords: stability
reviewed by: Bülent Karasözen
describes a project that uses: FreeFem++
MaRDI profile type: Publication
full work available at URL: jnma-2019-0108
OpenAlex ID: W3107557866
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DOI: 10.1515/JNMA-2019-0108
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description: scientific article; zbMATH DE number 7394126