A stabilized space-time discretization for the primitive equations in oceanography

Summary. In this work we introduce and analyze a space-time discretization for the Primitive Equations of the Ocean. We use a reduced formulation of these equations which only includes the (3D) horizontal velocity and the (2D) surface pressure (cf.19,20]). We use a semi-implicit Backward Euler scheme for the time discretization. The spatial discretization in each time step is carried out through a Penalty Stabilized Method. This allows to circumvent the use of pairs of spaces satisfying the inf-sup condition, thus attempting a large saving of degrees of freedom. We prove stability estimates, from which we deduce weak convergence in two steps : first in space to a semi-discretisation in time, and then in time to the continuous problem. Finally, we show a numerical test in a real-life application. Specifically, we simulate the wind-driven circulation in the Leman lake.Mathematics Subject Classification (2000): 65N30, 76M10, 86A05Revised version received July 1, 2002Research partially supported by Spanish Government Research Projects: MAR97-1055-C02-02, REN2000-1162-C02-01 MAR and REN2000-1168-C02-01 MAR