Compatible finite elements for terrain following meshes

We are presenting a new approach for compatible finite element discretisations for atmospheric flows on a terrain following mesh. In classical compatible finite element discretisations, the H(div)-velocity space involves the application of Piola transforms when mapping from a reference element to the physical element in order to guarantee normal continuity. In the case of a terrain following mesh, this causes an undesired coupling of the horizontal and vertical velocity components. We propose a new finite element space, that drops the Piola transform and introduce a hybridisable formulation with trace variables that are supported on horizontal cell faces in order to enforce the normal continuity of the velocity. Alongside the discrete formulation for various fluid equations we discuss solver approaches that are compatible with them and present our latest numerical and analytical results.