Error Analysis of a Continuous-Discontinuous Galerkin Finite Element Method for Generalized 2D Vorticity Dynamics

A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method for the generalized 2D vorticity dynamics equations which 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 discontinous Galerkin finite element method for the (potential) vorticity. Since this algorithm satisfies a number of invariants, such as energy and enstrophy conservation, it is a 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(Ω) obtained in a previous analysis to W r p (Ω) with r > 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.