This article surveys research on the application of compatible finite element
methods to large scale atmosphere and ocean simulation. Compatible finite
element methods extend Arakawa's C-grid finite difference scheme to the finite
element world. They are constructed from a discrete de Rham complex, which is a
sequence of finite element spaces which are linked by the operators of
differential calculus. The use of discrete de Rham complexes to solve partial
differential equations is well established, but in this article we focus on the
specifics of dynamical cores for simulating weather, oceans and climate. The
most important consequence of the discrete de Rham complex is the
Hodge-Helmholtz decomposition, which has been used to exclude the possibility
of several types of spurious oscillations from linear equations of geophysical
flow. This means that compatible finite element spaces provide a useful
framework for building dynamical cores. In this article we introduce the main
concepts of compatible finite element spaces, and discuss their wave
propagation properties. We survey some methods for discretising the transport
terms that arise in dynamical core equation systems, and provide some example
discretisations, briefly discussing their iterative solution. Then we focus on
the recent use of compatible finite element spaces in designing structure
preserving methods, surveying variational discretisations, Poisson bracket
discretisations, and consistent vorticity transport.Comment: correction of some typo
