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Title: Structure-preserving finite element methods for fluids
Author: Natale, Andrea
ISNI:       0000 0004 7427 7205
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2017
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Many fluid models share a common geometric structure which is usually ignored by the standard algorithms used in computer simulations. As a consequence, numerical solutions may often exhibit unphysical behaviour and fail to reproduce certain features of the governing equations, such as conservation laws or symmetries. In this thesis we use the framework of finite element exterior calculus to derive discretisations for fluid systems that preserve as much as possible of their geometric structure. The main part of this work is devoted to the discretisation of the incompressible Euler equations. We construct a finite-dimensional approximation of these equations by mimicking their variational derivation and preserving the particle relabelling symmetry. As a consequence, we obtain a finite element scheme that conserves energy and possesses a discrete version of Kelvin’s circulation theorem. The variational derivation also introduces a new form of energy-conserving upwind stabilisation in the discrete advection term. We find that this reproduces the energy backscatter which characterises two-dimensional turbulence; we study this behaviour numerically, by comparing our discretisation with other types of upwind schemes. In addition, we address a number of issues related to structure preservation in finite element discretisations, primarily motivated by applications in geophysical fluid dynamics. Specifically, we study the approximation properties of tensor product finite element spaces of differential forms and we show how to generalise these to manifolds. Moreover, we use similar spaces to characterise the convergence to zonal flows of time-averaged discrete approximations of incompressible flows on a rotating sphere.
Supervisor: Cotter, Colin Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral