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.
Highlights
Finite element methods have recently become a popular discretization approach for numerical weather prediction, mostly using spectral elements or discontinuous Galerkin methods (Bao et al, 2015; Brdar et al, 2013; Dennis et al, 2011; Fournier et al, 2004; Giraldo et al 2013; Kelly and Giraldo, 2012; Marras et al, 2013; Thomas and Loft, 2005); the use of these methods is reviewed in the work by Marras et al (2015)
Compatible finite element methods are built from finite element spaces that have differential operators that map from one space to another
They have not been considered much in the numerical weather prediction context, the lowest Raviart–Thomas element has been proposed for ocean modelling (Walters and Casulli, 1998), and the RT0 and lowest order Brezzi–Douglas–Marini element were both analysed in Rostand and Le Roux (2008) as part of the quest for a mixed finite element pair with good dispersion properties when applied to geophysical fluid dynamics
Summary
Finite element methods have recently become a popular discretization approach for numerical weather prediction, mostly using spectral elements or discontinuous Galerkin methods (Bao et al, 2015; Brdar et al, 2013; Dennis et al, 2011; Fournier et al, 2004; Giraldo et al 2013; Kelly and Giraldo, 2012; Marras et al, 2013; Thomas and Loft, 2005); the use of these methods is reviewed in the work by Marras et al (2015). Finite element methods provide the opportunity to use more general grids in numerical weather prediction, either to improve load balancing in massively parallel computation or to facilitate adaptive mesh refinement They allow the development of higher order discretizations. Compatible finite element methods are built from finite element spaces that have differential operators that map from one space to another They have a long history in both numerical analysis (see Boffi et al, 2013, for a summary of contributions) and applications, including aerodynamics, structural mechanics, electromagnetism and porous media flows.
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