Compatible finite element spaces for geophysical fluid dynamics

TL;DR

This paper reviews the use of compatible finite element spaces in geophysical fluid dynamics, highlighting their structure-preserving properties and recent advances in approximation and hydrostatic balance in 3D models.

Contribution

It provides a comprehensive survey of compatible finite element applications in geophysical fluid dynamics, including new results on 3D approximation and hydrostatic balance.

Findings

Compatible finite elements preserve key physical structures.

New approximation results for 3D on the sphere.

Enhanced understanding of hydrostatic balance properties.

Abstract

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.