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Finite Element Approximation of Hydrostatic Stokes Equations: Review and Tests

  • Francisco Guillén-González
  • J. Rafael Rodríguez-Galván
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)

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 Some other unpublished numerical experiments are also presented here, in singular 2D domains and in realistic 3D domains (Gibraltar Strait).

Keywords

Depth Function Unstructured Mesh Finite Element Approximation Mixed Finite Element Error Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The work was partially supported by MINECO grant MTM2012-32325 with the participation of FEDER. We would like to thank the referees for providing useful comments which served to improve the paper.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Francisco Guillén-González
    • 1
  • J. Rafael Rodríguez-Galván
    • 2
  1. 1.Dpto. Ecuaciones Diferenciales y Análisis Numérico and IMUSUniversidad de SevillaSevillaSpain
  2. 2.Dpto. de MatemáticasUniversidad de Cádiz, Facultad de CienciasPuerto RealSpain

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