The compatible finite element method has recently gained an increased interest in numerical weather prediction, as it allows for higher order discretisations and more general meshes, thus avoiding the parallel computing issues associated with grid poles. Further, it can be seen as a finite element extension of the Arakawa finite difference C grid. A dynamical core based on it is currently in development at the UK Met Office, due to replace the current finite difference latitude longitude grid discretisation.
In this thesis, we focus on compatible finite element discretisations that are energy conserving, which is an important property in particular for climate simulations. This is achieved using a Hamiltonian framework, where energy conservation is expressed via the antisymmetry of a Poisson bracket that underlies the governing equations. In order to achieve an improved field development, we newly incorporate into the Poisson bracket Discontinuous Galerkin upwind and Streamline Upwind Petrov Galerkin stabilisation methods for the thermal, density related, and vorticity field equations. The energy conserving property is validated by coupling the upwind-stabilised spatial discretisations to an energy conserving time discretisation. Further, the discretisations are demonstrated to lead to an improved field development with respect to stability for the 2D and 3D dry compressible Euler, rotating shallow water, and thermal rotating shallow water equations.