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On Stability of Discontinuous Galerkin Approximations to Anisotropic Stokes Equations

  • Francisco Guillén-González
  • María Victoria Redondo-Neble
  • José Rafael Rodríguez-GalvánEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 18)

Abstract

This work delves into the numerical approximation of Anisotropic Stokes equations (with small vertical diffusion coefficient), which is a generalization of the Hydrostatic Stokes equations (with zero vertical diffusion). It is known that the Ladyzhenskaya-Babuška-Brezzi condition is not sufficient to stabilize usual finite elements approximations, because a new stability condition appears. Here we extend to the Anisotropic Stokes equations the new approach given in Guillén González et al. (On stability of discontinuous galerkin approximations to the hydrostatic Stokes equations, 2018, submitted) for the Hydrostatic case. This approach is a symmetric interior penalty discontinuous Galerkin method (SIP DG) with adequate stability terms, approximating both velocity and pressure in the same Finite Element (FE) space (\(\ensuremath {\mathcal {P}_{k}}\)-discontinuous). Stability and well-posedness of this method is proven. Finally, we show some numerical tests in agreement with our numerical analysis.

Keywords

Stokes equations Anisotropic viscosity Discontinuous Galerkin Finite elements Fluid mechanics Stability 

Notes

Acknowledgements

The first author has been partially financed by the MINECO grant MTM2015-69875-P (Spain) with the participation of FEDER. The second and third authors are also partially supported by the research group FQM-315 of Junta de Andalucía.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Francisco Guillén-González
    • 1
  • María Victoria Redondo-Neble
    • 2
  • José Rafael Rodríguez-Galván
    • 2
    Email author
  1. 1.Departamento de Ecuaciones Diferenciales y Análisis NuméricoIMUS, Universidad de SevillaSevillaSpain
  2. 2.Departamento de Matemáticas, Universidad de CàdizCàdizSpain

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