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:文章调查对大型大气和海洋模拟应用相容的有限元素方法的研究。 兼容的有限元素方法将Arakawa的C- 电离电网的有限差异办法延伸至有限元素世界。 它们是由离散的Rham综合体建造的, 这是一种由不同微积分操作者连接的有限元素空间的序列。 使用离散的 de Rham综合体解决部分差异方程式的有用框架已经确立, 但在本条中, 我们侧重于模拟天气、 海洋和气候的动态核心的具体特性。 离散的Rham综合体的最重要后果是Hodge- Helmholtz 脱腐化, 用于排除地球物理流动线性方程式中若干种类的刺激性振荡的可能性。 这意味着兼容的有限元素空间为建立部分差异方程式提供了有用的框架。 在本条中, 我们介绍了兼容的有限元素空间的主要概念, 并讨论了其波波传播特性。 我们调查了在动态核心方程式系统中产生的运输术语的离散化方法, 并且提供了一些实例的离散化、 离散的离析性结构, 、 保持其最小的移动性结构 、 分析 、 的最小的流流变异化的系统。</s>