Morley finite element methods for the stationary quasi-geostrophic equation

Abstract

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf–sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of the existence of an isolated solution to the linearized problem and Banach fixed point argument. One principal tool employed is to exploit the enrichment operator from nonconforming space to H02(Ω). Thereby, not only smallness assumption on data is avoided, but also optimal error estimates in H2- and H1-norms are proved under minimal regularity condition on the exact solution. Then, for the nonlinear discrete system, the Newton method is applied, which is shown to preserve local quadratic convergence. Moreover, a posteriori error estimator for an adaptive algorithm is derived. Finally, several numerical experiments with a benchmark problem are considered to confirm our theoretical findings.