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Published in:
2019 | OriginalPaper | Chapter
On Stability of Discontinuous Galerkin Approximations to Anisotropic Stokes Equations
Authors : Francisco Guillén-González, María Victoria Redondo-Neble, José Rafael Rodríguez-Galván
Published in: Recent Advances in Differential Equations and Applications
Publisher: Springer International Publishing
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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.