Stabilized Schemes for the Hydrostatic Stokes Equations

Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes system or primitive equations of the ocean. It is known that the stability of the mixed formulation approximation for primitive equations requires the well-known Ladyzhenskaya--Babuska--Brezzi condition related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical velocity [F. Guillen-Gonzalez and J. R. Rodriguez-Galvan, Numer. Math., 130 (2015), pp. 225--256]. The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to the primitive equations and some error estimates are provided using Taylor--Hood $\mathcal{P}_2$--$\mathcal{P}_1$ or minielement $(\mathcal{P}_1+\text{bubble})$--$\mathcal{P}_1$ FE approximations, showing the optimal convergence rate in the $\mathcal{P}_2$--$\mathcal{P}_1$ case. These results are also extended to the anisotropic...

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