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

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