Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.

We propose a finite element discretisation approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite element method that arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite element H(div) vector fields are identified with advection operators; this is the first successful extension of the structure-preserving discretisation of to the finite element setting. The resulting algorithm coincides with the energyconserving scheme presented in Guzmán et al. (2016). Through the variational derivation, we discover that it also satisfies a discrete analogous of Kelvin's circulation theorem. Further, we propose an upwind-stabilised version of the scheme which dissipates enstrophy whilst preserving energy conservation and the discrete Kelvin's theorem. We prove error estimates for this version of the scheme, and we study its behaviour through numerical tests.One of the most recent approaches is the one associated to the school of Brenier [8,9], see, e.g., the discretisations proposed in [17,28]. Such methods are based on the reformulation of the variational principle underlying the incompressible Euler equations into an optimal transport problem. This leads to the definition of the notion of generalised incompressible flow, in which the motion of the fluid particles is defined in a probabilistic sense. The methods in [17,28] aim at producing approximations of the flow map in this setting. In particular, they construct minimizing geodesics between discrete measure preserving maps transporting the particle mass distributions. Moreover, they establish convergence to classical solutions of the Euler equations.In this paper we follow a different route, which is inspired by the work of Pavlov et al. [30,31]. In this case, the discrete flow maps are discretised by considering their action on piecewise constant scalar functions intended as right composition. However, such maps are not constructed explicitly and the final algorithm is fully Eulerian, i.e. one approximates the evolution of the velocity field at fixed positions in space, without following each fluid particle. One key ingredient to achieve this is the identification of vector fields with discrete Lie derivatives, i.e. advection operators, also acting on piecewise constant scalar functions. Owing to this identification, one can reduce the variational principle governing the dynamics from the space of discrete flow maps to the one of discrete velocity fields, i.e. from a Lagrangian (material) description to an Eulerian (spatial) one. The final algorithm can be interpreted as an advection problem for the velocity one-form by means of another discrete Lie derivative operator, defined consistently with the Discrete Exterior Calculus (DEC) formalism [15]. Desbrun et al. [14] extended this promising idea to generate a variational numerical scheme for rotating stratified fluid models. Both the algorithms presented in [31] and [14] ca...

We analyze the multiscale properties of energy‐conserving upwind‐stabilized finite‐element discretizations of the two‐dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretization introduced by Natale and Cotter and the Streamline Upwind/Petrov–Galerkin (SUPG) discretization of the vorticity advection equation. Such discretizations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non‐local energy backscatter that characterizes two‐dimensional turbulent flows.

Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes to discretize these problems preserving their variational structure and obtaining second order accuracy in space. The choice of the discrete solver plays an important role in designing these schemes for robustness purposes. We present two applications to test the scheme and show its order of convergence.