  A new numerical method is presented for solving the rotating shallow water
equations on a rotating sphere using quasi-uniform polygonal meshes. The method
uses special families of finite element function spaces to mimic key
mathematical properties of the continuous equations and thereby capture several
desirable physical properties related to balance and conservation. The method
relies on two novel features. The first is the use of compound finite elements
to provide suitable finite element spaces on general polygonal meshes. The
second is the use of dual finite element spaces on the dual of the original
mesh, along with suitably defined discrete Hodge star operators to map between
the primal and dual meshes, enabling the use of a finite volume scheme on the
dual mesh to compute potential vorticity fluxes. The resulting method has the
same mimetic properties as a finite volume method presented previously, but is
more accurate on a number of standard test cases.
