A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is L2-stable in the enstrophy norm. A detailed error analysis has been carried out which imposes only minimal smoothness requirements on the vorticity field. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation appears to conserve energy and is L2-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.