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.

翻译：本文综述了相容有限元方法在大尺度大气与海洋模拟中的应用研究。相容有限元方法将荒川C网格有限差分格式推广至有限元领域，其构建基础为离散德·拉姆复形——一个通过微分算子链接的有限元空间序列。利用离散德·拉姆复形求解偏微分方程已是成熟技术，但本文聚焦于气象、海洋及气候模拟中动力核心的特殊需求。离散德·拉姆复形最重要的结果是霍奇-亥姆霍兹分解，该分解已被用于排除地球物理流动线性方程中多种虚假振荡的可能性。这意味着相容有限元空间为构建动力核心提供了有效框架。本文首先介绍相容有限元空间的主要概念，并讨论其波动传播特性。随后综述动力核心方程系统中传输项离散化的若干方法，提供部分离散化示例并简要讨论其迭代求解方案。最后聚焦近年来相容有限元空间在保结构方法设计中的应用，系统梳理变分离散化、泊松括号离散化及一致性涡量输运等研究方向。