Acta Numerica (2023), pp. 291–393 Printed in the United Kingdom

doi:10.1017/S0962492923000028

Compatible finite element methods for

geophysical fluid dynamics

Colin J. Cotter

Department of Mathematics, Imperial College London,

South Kensington Campus, London SW7 2AZ, UK

E-mail: [email protected]

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.

2020 Mathematics Subject Classification:

65M60, 65P10, 65Z05, 76U60, 86A05, 86A08, 86A10

© The Author(s), 2023. Published by Cambridge University Press.

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 cite

First of all, the term 'fluid mechanics' is more commonly used than 'fluid mechanic'

The Ritz method, the method of weighted residuals (MWR), and variational methods are all closely related approaches for approximating solutions to partial differential equations (PDEs) that arise in various fields, including fluid mechanics. They share the common goal of finding an approximate solution that minimizes a residual or energy functional.

In the context of fluid mechanics, these methods are commonly used to approximate solutions to the Navier-Stokes equations, which govern the flow of viscous fluids. The choice of method depends on the specific problem and the desired level of accuracy.

All right, let's have a quick review:

The Ritz method, also known as the Rayleigh-Ritz method, is a general approach for solving variational problems. It involves approximating the unknown solution using a trial function that is expanded in terms of a set of basis functions. The residual is then formed by substituting the trial function into the governing PDE and multiplying by a weighting function. The solution is found by minimizing the residual over the set of admissible trial functions.

The method of weighted residuals (MWR) is a broader class of numerical methods that encompasses the Ritz method and other approaches. It involves discretizing the governing PDE by introducing a trial function and a weighting function. The residual is then formed by substituting the trial function into the PDE and multiplying by the weighting function. The solution is found by enforcing the residual to be zero at a set of collocation points or by integrating the residual over the domain of the problem.

Variational methods are based on the principle that the true solution of a physical system minimizes a certain energy functional. This energy functional can be derived from the governing PDEs using Hamilton's principle or other variational principles. The solution is then found by minimizing the energy functional over the set of admissible trial functions.