**A (Dis)Continuous Finite Element Model for Generalized 2D Vorticity
Dynamics**

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.

*Onno Bokhove*

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