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Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green–Naghdi Equations

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Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green{Naghdi Equations by Adam Morgan A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Applied Mathematics Waterloo, Ontario, Canada, 2018 c Adam Morgan 2018 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract The study of finite element methods for the numerical solution of differential equations is one of the gems of modern mathematics, boasting rigorous analytical foundations as well as unambiguously useful scientific applications. Over the past twenty years, several researchers in scientific computing have realized that concepts from homological algebra and differential topology play a vital role in the theory of finite element methods. Finite element exterior calculus is a theoretical framework created to clarify some of the relationships between finite elements, algebra, geometry, and topology. The goal of this thesis is to provide an introduction to the theory of finite element exterior calculus, and to illustrate some applications of this theory to the design of mixed finite element methods for problems in geophysical fluid dynamics. The presentation is divided into two parts. Part 1 is intended to serve as a self{contained introduction to finite element exterior calculus, with particular emphasis on its topological aspects. Starting from the basics of calculus on manifolds, I go on to describe Sobolev spaces of differential forms and the general theory of Hilbert complexes. Then, I explain how the notion of cohomology connects Hilbert complexes to topology. From there, I discuss the construction of finite element spaces and the proof that special choices of finite element spaces can be used to ensure that the cohomological properties of a particular problem are preserved during discretization. In Part 2, finite element exterior calculus is applied to derive mixed finite element methods for the Green{Naghdi equations (GN). The GN extend the more well{known shallow water equations to the regime of non–infinitesimal aspect ratio, thus allowing for some non{hydrostatic effects. I prove that, using the mixed formulation of the linearized GN, approximations of balanced flows remain steady. Additionally, one of the finite element methods presented for the fully nonlinear GN provably conserves mass, vorticity, and energy at the semi{discrete level. Several computational test cases are presented to assess the practical performance of the numerical methods, including a collision between solitary waves, the motion of solitary waves over variable bottom topography, and the breakdown of an unstable balanced state. iii Acknowledgements First, I would like to thank my supervisors, Francis Poulin and Benoit Charbonneau. In my time working with them, they have both displayed incredible patience, as evidenced by them actually reading this kaiju of a thesis. My project involved dragging both of them outside of the domains of their usual research, but they always showed enthusiasm and support. Francis' advice on coding has been instrumental to the success of my numerical routines, and I cannot adequately express how much I appreciate Benoit's amazing editorial skills. I would also like to thank the other members of my degree committee, Kevin Lamb and Sander Rhebergen, for helpful comments and discussions. The study of finite element methods has formed a substantial part of my intellectual life for the past two years, and I would like to thank everyone who has helped shape my understanding of this discipline. First, Colin Cotter and Jemma Shipton have been enormously helpful in my quest to learn Firedrake. On the theoretical side of finite elements, I have learned a great deal during discussions with Keegan Kirk, whose devotion to and passion for numerical analysis is very inspiring. Thanks are also due to Sander Rhebergen (again) and Tamas Horvath, who were always happy to help me whenever I barged in on them. I have had the privilege to work closely with many talented graduate students during my time at Waterloo, including Leon Avery (who shared with me the pains of learning to work with finite element software), Supranta Sarma Boruah (who shared with me the pains of learning canonical perturbation theory), Fabian Germ (who directed me to Cauchy's very interesting life story), Parham Hamidi (with whom I had many fruitful discussions on algebraic geometry), Anthony McCormick (my spirit{brother, the first person I met who cared about infinite–dimensional manifolds as much as I do), Jeff Samuelson (\to thine own self be true"), and Ben Storer (the Python wizard). I also would like to thank all of my friends back in Edmonton for moral support, especially Mitchell Taylor, who was always willing to listen to my complaining. I owe a great debt to many of the wonderful teachers I have had over the past six years. First, I would like to thank Vincent Bouchard and Vakhtang Putkaradze for cultivating my initial interest in geometry and physics. I am also grateful to Gordon Swaters, from whom I learned to properly think like a theoretical physicist. I owe Manish Patnaik for sharpening my mathematical maturity and writing skills. At Waterloo, I was very fortunate to learn from Spiro Karigiannis, who taught me to love all of the gritty details of differential geometry, and Matt Satriano, who has the honour of being the person who got me to care about ring theory. Lastly, I cannot thank Xinwei Yu enough for baptizing his Math 217/317 class by fire; he is a continuing inspiration who taught me some of the most valuable lessons about mathematics and life. Finally, I must extend gratitude to my family for their encouragement and patience. Most importantly, I would like to thank my partner Kristen Cote, without whom I would certainly be living in a dumpster iv behind a 7−11. Her input on both the written and visual components of this thesis has been immeasurably helpful. v Table of Contents Author's Declaration ii Abstract iii Acknowledgements iv List of Figures x 1 Introduction 1 1.1 Prelude: The Shape of Numerical PDE Theory......................... 1 1.2 Goals of this Thesis ........................................ 2 1.3 Outline of Chapters ........................................ 3 1.4 Notation............................................... 4 I An Introduction to Finite Element Exterior Calculus6 2 Calculus on Smooth Manifolds: A User's Manual7 2.1 Manifolds, Vectors, and Tensors.................................. 8 2.2 Basics of Differential Forms.................................... 16 2.3 Orientability and Hodge Duality ................................. 20 2.4 The Exterior Derivative ...................................... 22 2.5 Integration of Differential Forms ................................. 24 3 Hilbert Spaces of Differential Forms 27 3.1 Lipschitz Manifolds......................................... 28 3.2 The Spaces L2Λk .......................................... 30 3.3 Weak Derivatives and the Spaces HΛk .............................. 35 3.4 The Trace Theorem ........................................ 38 vi 4 Hilbert Complexes 39 4.1 Some Facts from Functional Analysis............................... 40 4.2 Hilbert Complexes: Basic Definitions............................... 44 4.3 Cohomology............................................. 47 4.4 Harmonic Forms and the Hodge{Helmholtz Decomposition .................. 48 4.5 The Compactness Property .................................... 52 5 Triangulations and Topology 56 5.1 Simplices and Simplicial Complexes ............................... 57 5.2 Homology of Simplicial Complexes................................ 62 5.3 de Rham's Theorem: Bridging Analysis and Topology ..................... 70 6 Generalities on Finite Element Methods 73 6.1 Introduction to Galerkin Methods and Finite Element Methods................ 74 6.2 Finite Elements........................................... 77 6.3 Local{to{Global Maps and Finite Element Spaces ....................... 81 6.4 Characterization of d{Conforming Finite Element Spaces ................... 84 7 Polynomial Differential Forms on Simplices 87 7.1 Binomial Coefficient Identities................................... 87 k 7.2 PrΛ ................................................. 89 − k 7.3 The Koszul Operator and Pr Λ ................................. 90 8 Construction of Finite Element Spaces 99 0 8.1 The Lagrange Finite Element Space PrΛ (Th) .........................100 8.2 Vanishing Lemmas.........................................105 − k 8.3 The Trimmed Finite Element Space Pr Λ (Th).........................107 k 8.4 The Regular Finite Element Space PrΛ (Th) ..........................115 9 Cohomology of Finite Element Spaces 120 9.1 Approximating Hilbert Complexes ................................121 9.2 Projections onto Finite Element de Rham Complexes......................128 9.3 de Rham Theory `ala Whitney ..................................131 9.4 Two Big Theorems.........................................138 9.5 The Periodic Table of the Finite Elements............................139 vii II Applications to the Numerical Solution of the Green{Naghdi Equa- tions 144 10 The Green{Naghdi Equations 145 10.1 Derivation of the Green{Naghdi Equations............................146 10.2 Adding the Effects of Rotation ..................................152
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