Error analysis of a continuous-discontinous Galerkin finite element model for generalized 2D vorticity dynamics

Abstract. A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin ��?nite element method for the generalized 2D vorticity dynamics equations. These equations describe several types of geophysical flows, including the Euler equations. The algorithm consists of a continuous Galerkin ��?nite element method for the stream function and a discontinous Galerkin ��?nite element method for the (potential) vorticity. Since this algorithm satis��?es a number of invariants, such as energy and enstrophy conservation, it is possible to provide detailed error estimates for this non-linear problem. The main result of the analysis is a reduction in the smoothness requirements on the vorticity field from $H^2(\Omega),$ obtained in a previous analysis, to $W_p^r(\Omega)$ with $r>\frac{1}{p}$ and $p>2.$ In addition, sharper estimates for the dependence of the error on time and numerical examples on a model problem are provided.