Abstract
A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method for the generalized two-dimensional vorticity dynamics equations. These equations describe several types of geophysical flows, including the Euler equations. The algorithm consists of a continuous Galerkin finite element method for the stream function and a discontinuous Galerkin finite element method for the (potential) vorticity. Since this algorithm satisfies a number of invariants, such as energy and enstrophy conservation, it is possible to provide detailed error estimates for this nonlinear 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 $W^r_p(\Omega)$ 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.
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