地球流体动力学的相容有限元法
Compatible finite element methods for geophysical fluid dynamics
摘要：
本文综述了相容有限元法在大尺度大气和海洋模拟中的应用研究。相容有限元法将荒川的C-网格有限差分格式推广到有限元领域。它们是从离散的de Rham复形构造的，该复形是由微分学的算子连接的有限元空间的序列。使用离散的德拉姆复形来求解偏微分方程是公认的，但在这篇文章中，我们专注于模拟天气，海洋和气候的动力核心的细节。离散的de Rham复形的最重要的结果是Hodge-Helmholtz分解，它已被用来从地球物理流的线性方程中排除几种寄生振荡的可能性。这意味着相容的有限元空间为构建动态核提供了一个有用的框架。本文介绍了相容有限元空间的主要概念，并讨论了它们的波传播性质。我们调查了一些方法离散的动力学核心方程系统中出现的运输条款，并提供了一些例子离散，简要讨论他们的迭代解。然后，我们专注于最近使用的兼容有限元空间在设计结构保持方法，调查变分离散，泊松括号离散，和一致的涡度输运。
英文摘要：
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.
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