The quasi-geostrophic equations (QGE) are a model of large-scale ocean flows. We consider a pure stream function formulation and cite results for optimal error estimates for finding approximate solutions with the finite element method. We examine both the time dependent and steady-state versions of the equations. Numerical experiments verify the error estimates. We examine the Argyris finite element and derive the transformation matrix necessary to perform calculations on the reference triangle. We use the Argyris element because it is a high-order, conforming finite element for fourth order problems. In order to increase computational efficiency, we consider a two-level method to linearize the system of equations. This allows us to solve a small, nonlinear system and then use the result to linearize a larger system. / Master of Science