Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations

This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O(k+hl)O(k+hl) for l=1l=1 or l=2l=2 considering either first or second order finite-element approximations (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≤Ch2.