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Compatible finite element methods for geophysical fluid dynamics

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical problems in dynamical systems Geophysics Geophysics (general)

Published online by Cambridge University Press:  11 May 2023

Colin J. Cotter Open the ORCID record for Colin J. Cotter [Opens in a new window]
Colin J. Cotter*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK E-mail: colin.cotter@imperial.ac.uk
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Abstract

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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.

MSC classification

Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65P10: Hamiltonian systems including symplectic integrators 65Z05: Applications to physics 86A05: Hydrology, hydrography, oceanography 86A10: Meteorology and atmospheric physics
Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 291 - 393
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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