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。其构建基于离散de Rham复形，该复形是由微分算子连接的一系列有限元空间构成的序列。利用离散de Rham复形求解偏微分方程已有完善的理论基础，但本文重点关注将其应用于模拟天气、海洋及气候的动力核心的特殊性。离散de Rham复形最重要的结果是Hodge-Helmholtz分解，该分解被用于排除地球物理流动线性方程中数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。本文介绍相容有限元空间的主要概念，并讨论其波动传播特性。我们综述了动力核心方程组中平流项离散化的若干方法，并列举了一些离散化范例，简要讨论了其迭代求解方式。最后重点阐述近年来相容有限元空间在保结构方法设计中的应用，综述了变分离散化、泊松括号离散化及一致性涡量输运的研究进展。