Convergence and error estimates of a viscosity-splitting finite-element scheme for the Primitive Equations

The purpose of this paper is the numerical analysis of a first order fractional-step time-scheme, using decomposition of theviscosity, and “inf-sup” stable finite element space-approximations for the Primitive Equations of the Ocean. The aim of the paper is twofold. Firstly, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Secondly, optimal error estimates for velocity and pressure are provided of order $O(k+h^l)$ for $l=1$ or $l=2$ when either first or second order finite-element approximations are considered ($k$ and $h$ being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint $k\le h^2$.