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Compatible finite element methods for geophysical fluid dynamics
Acta Numerica ( IF 14.2 ) Pub Date : 2023-05-11 , DOI: 10.1017/s0962492923000028
Colin J. Cotter

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 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 discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

中文翻译:

地球物理流体动力学的兼容有限元方法

本文综述了兼容有限元方法在大尺度大气和海洋模拟中的应用研究。兼容的有限元方法将 Arakawa 的 C 网格有限差分格式扩展到有限元世界。它们由离散的 de Rham 复形构成,该复形是由微分运算符连接的一系列有限元空间。使用离散 de Rham 复形来求解偏微分方程已经很成熟,但在本文中,我们将重点关注用于模拟天气、海洋和气候的动力学核心的细节。离散 de Rham 复形最重要的结果是 Hodge-Helmholtz 分解,它已被用于从地球物理流动的线性方程中排除几种类型的虚假振荡的可能性。这意味着兼容的有限元空间为构建动力核心提供了一个有用的框架。在本文中,我们介绍了兼容有限元空间的主要概念,并讨论了它们的波传播特性。我们调查了一些离散化动力核心方程系统中出现的传输项的方法,并提供了一些离散化示例,并简要讨论了它们的迭代解。然后我们关注最近在设计结构保持方法、测量变分离散化、泊松括号离散化和一致涡量传输中使用兼容的有限元空间。并讨论它们的波传播特性。我们调查了一些离散化动力核心方程系统中出现的传输项的方法,并提供了一些离散化示例,并简要讨论了它们的迭代解。然后我们关注最近在设计结构保持方法、测量变分离散化、泊松括号离散化和一致涡量传输中使用兼容的有限元空间。并讨论它们的波传播特性。我们调查了一些离散化动力核心方程系统中出现的传输项的方法,并提供了一些离散化示例,并简要讨论了它们的迭代解。然后我们关注最近在设计结构保持方法、测量变分离散化、泊松括号离散化和一致涡量传输中使用兼容的有限元空间。
更新日期:2023-05-11
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