Finite element approximation to the non-stationary quasi-geostrophic equation

In this paper, C 1 -conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation model. In its first part, some new regularity results are derived, which show an exponential decay property when the wind shear stress is zero or exponentially decaying. Moreover, when the wind shear stress is independent of time, the existence of an attractor is established. In its second part, finite element methods are applied in the spatial direction, and for the resulting semi-discrete scheme, the exponential decay property and the existence of a discrete attractor are proved. By introducing an intermediate solution of a discrete linearized problem, optimal error estimates are derived both for smooth and non-smooth solutions. Based on the backward Euler method, a completely discrete scheme is obtained and uniform in time a priori estimates are established. Moreover, the existence of a discrete solution is proved by appealing to a variant of the Brouwer fixed point theorem, and then, an optimal error estimate is derived. Finally, several computational experiments with benchmark problems are conducted to confirm our theoretical findings.