A compatible and conservative spectral element method on unstructured grids

We derive a formulation of the spectral element method which is compatible on very general unstructured three-dimensional grids. Here compatible means that the method retains discrete analogs of several key properties of the divergence, gradient and curl operators: the divergence and gradient are anti-adjoints (the negative transpose) of each other, the curl is self-adjoint and annihilates the gradient operator, and the divergence annihilates the curl. The adjoint relations hold globally, and at the element level with the inclusion of a natural discrete element boundary flux term. We then discretize the shallow-water equations on the sphere using the cubed–sphere grid and show that compatibility allows us to locally conserve mass, energy and potential vorticity. Conservation is obtained without requiring the equations to be in conservation form. The conservation is exact assuming exact time integration.