Colloquium - Colin Cotter - Compatible Finite Element Methods for Geophysical Fluid Dynamics
Title: Compatible Finite Element Methods for Geophysical Fluid Dynamics
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. Compatible finite element methods use combinations of finite element spaces chosen so that differential operators such as div, grad and curl map from one space to another. These methods, under various different names, have long been used in several applications e.g. porous media, but their use in geophysical fluid dynamics models for atmosphere and ocean flow is more recent. In this setting, the compatible structure leads to the exact preservation of geophysical balances in the linear and nonlinear setting, leading to compatible finite methods being adopted by the Met Office for their next generation atmosphere dynamical core. The structure also leads to energy conserving formulations when combined with a discrete Poisson bracket structure; energy can be conserved even in the presence of upwinding. I will 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, and will discuss scalable solution strategies to the implicit equation systems that arise through this framework.
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