Compatible finite element methods for geophysical fluid dynamics

TL;DR

This paper surveys the use of compatible finite element methods in large-scale atmospheric and oceanic simulations, highlighting their structure-preserving properties and advantages in reducing spurious oscillations.

Contribution

It introduces the main concepts of compatible finite element spaces and discusses their application in designing structure-preserving methods for geophysical fluid dynamics.

Findings

Compatible finite element spaces prevent spurious oscillations in linear geophysical flow equations.

They enable structure-preserving discretizations like variational and Poisson bracket methods.

These methods improve the accuracy and stability of dynamical core simulations.

Abstract

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.