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

TL;DR

This paper analyzes finite element methods for the non-stationary quasi-geostrophic equation, establishing regularity, decay properties, attractors, and optimal error estimates, supported by computational experiments.

Contribution

It introduces new regularity results, proves decay and attractor existence, and derives optimal error estimates for finite element approximations of the equation.

Findings

Exponential decay property established

Existence of discrete attractor proven

Optimal error estimates derived

Abstract

In this paper, C1-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 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. Based on 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, optimal error estimate is derived. Finally, several computational experiments with benchmark problems are conducted to confirm our theoretical findings.