A key part of numerical weather prediction is the simulation of the partial differential equations governing atmospheric flow over the Earth's surface. This is typically performed on supercomputers at national and international centres around the world. In the last decade, there has been a relative plateau in single-core computing performance. Running ever-finer forecasting models has necessitated the use of ever-larger numbers of CPU cores. Several current forecasting models, including those

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... voured by the Met Office, use an underlying latitude--longitude grid. This facilitates the development of finite difference discretisations with favourable numerical properties. However, such models are inherently unable to make efficient use of large numbers of processors, as a result of the excessive concentration of gridpoints in the vicinity of the poles. A certain class of mixed finite element methods have recently been proposed in order to obtain favourable numerical properties on an arbitrary -- in particular, quasi-uniform -- mesh. This thesis supports the proposition that such finite element methods, which we label "compatible", or "mimetic", are suitable for discretising the equations used in an atmospheric dynamical core. We firstly show promising results applying these methods to the nonlinear rotating shallow-water equations. We then develop sophisticated tensor product finite elements for use in 3D. Finally, we give a discretisation for the fully-compressible 3D equations.