Stabilized schemes for the hydrostatic Stokes equations

Abstract

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. Guillén-González and J. R. Rodríguez-Galván, 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 P2–P1 or minielement (P1 +bubble)–P1 FE approximations, showing the optimal convergence rate in the P2–P1 case. These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand, by adding another residual term to the continuity equation, a better approximation of the vertical derivative of pressure is obtained. In this case, stability and error estimates including this better approximation are deduced, where optimal convergence rate is deduced in the (P1 +bubble)–P1 case. Finally, some numerical experiments are presented supporting previous results.