Numerical solution of the Primitive Equations of the ocean by the Orthogonal Sub-Scales VMS method

This paper deals with the development of efficient numerical solvers for the Primitive Equations of the ocean. We consider weak solutions of a reduced model that includes the horizontal velocity and the surface pressure. We derive the numerical approximation of this model by the Orthogonal Sub-Scales (OSS) method via finite elements discretization. We perform the numerical analysis of this discretization (stability, convergence, error estimates) for a linearized model, obtaining optimal error estimates for 2D flows. This analysis is based upon a specific inf–sup condition for the OSS discretization. We also perform some numerical tests for the non-linear Primitive Equations, that confirm the theoretical convergence order expectations, and show an improved convergence with respect to standard mixed methods.