This paper presents a conforming finite element discretization of the streamfunction formulation of the one-layer stationary quasi-geostrophic equations, which are a commonly used model for the large scale wind-driven ocean circulation. Optimal error estimates for this finite element discretization with the Argyris element are derived. To the best of the authors’ knowledge, these represent the first optimal error estimates for the finite element discretization of the quasi-geostrophic equations. Numerical tests for the finite element discretization of the quasi-geostrophic equations and two of its standard simplifications (the linear Stommel model and the linear Stommel–Munk model) are carried out. By benchmarking the numerical results against those in the published literature, we conclude that our finite element discretization is accurate. Furthermore, the numerical results have the same convergence rates as those predicted by the theoretical error estimates.@en

In this paper we proposed a two-level finite element discretization of the nonlinear stationary quasi-geostrophic equations, which model the wind driven large scale ocean circulation. Optimal error estimates for the two-level finite element discretization were derived. Numerical experiments for the two-level algorithm with the Argyris finite element were also carried out. The numerical results verified the theoretical error estimates and showed that, for the appropriate scaling between the coarse and fine mesh sizes, the two-level algorithm significantly decreases the computational time of the standard one-level algorithm.@en