  A mixed mimetic spectral element method is applied to solve the rotating
shallow water equations. The mixed method uses the recently developed spectral
element histopolation functions, which exactly satisfy the fundamental theorem
of calculus with respect to the standard Lagrange basis functions in one
dimension. These are used to construct tensor product solution spaces which
satisfy the generalized Stokes theorem, as well as the annihilation of the
gradient operator by the curl and the curl by the divergence. This allows for
the exact conservation of first order moments (mass, vorticity), as well as
quadratic moments (energy, potential enstrophy), subject to the truncation
error of the time stepping scheme. The continuity equation is solved in the
strong form, such that mass conservation holds point wise, while the momentum
equation is solved in the weak form such that vorticity is globally conserved.
While mass, vorticity and energy conservation hold for any quadrature rule,
potential enstrophy conservation is dependent on exact spatial integration. The
method possesses a weak form statement of geostrophic balance due to the
compatible nature of the solution spaces and arbitrarily high order spatial
error convergence.
