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.
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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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Guillén-González, F., Redondo-Neble, M.V., Rodríguez-Galván, J.R. (2019). On Stability of Discontinuous Galerkin Approximations to Anisotropic Stokes Equations. In: García Guirao, J., Murillo Hernández, J., Periago Esparza, F. (eds) Recent Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-00341-8_13
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DOI: https://doi.org/10.1007/978-3-030-00341-8_13
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