UNIVERSITY OF CALIFORNIA, SAN DIEGO Computation and Visualization of Geometric Partial Differential Equations A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Christopher L. Tiee Committee in charge: Professor Michael Holst, Chair Professor Bennett Chow Professor Ken Intriligator Professor Xanthippi Markenscoff Professor Jeff Rabin 2015 Copyright Christopher L. Tiee, 2015 All rights reserved. The dissertation of Christopher L. Tiee is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2015 iii DEDICATION To my grandfathers, Henry Hung-yeh Tiee, Ph. D. and Jack Fulbeck, Ph. D. for their inspiration. iv EPIGRAPH L’étude approfondie de la nature est la source la plus féconde des découvertes mathématiques. [Profound study of nature is the most fertile source of mathematical discoveries.] —Joseph Fourier, Theorie Analytique de la Chaleur v TABLE OF CONTENTS Signature Page........................................ iii Dedication.......................................... iv Epigraph...........................................v Table of Contents...................................... vi List of Figures........................................ ix List of Tables......................................... xi List of Supplementary Files................................ xii Acknowledgements..................................... xiii Vita.............................................. xv Abstract of the Dissertation................................ xvi Introduction.........................................1 0.1 The Main Problem: its Motivation and Antecedents......2 0.2 Part-by-Part Summary.......................7 I Background 10 Chapter 1 Boundary Value Problems........................ 11 1.1 Differential Forms.......................... 13 1.2 Integration of Differential Forms and Hodge Duality..... 19 1.3 Sobolev Spaces of Differential Forms............... 30 1.4 The Extended Trace Theorem................... 36 1.5 Boundary Value Problems with the Hodge Laplacian..... 42 1.6 The Hilbert Space Setting for Elliptic Problems......... 56 1.6.1 Recasting in terms of Sobolev Spaces.......... 57 1.6.2 The General Elliptic Problem............... 63 1.7 The Theory of Weak Solutions................... 69 1.7.1 The Lax-Milgram Theorem................ 71 1.7.2 Basic Existence Theorems................. 72 1.8 Hilbert Complexes......................... 76 1.9 Evolutionary Partial Differential Equations........... 87 1.9.1 Motivation: The Heat Equation............. 88 1.9.2 Bochner Spaces....................... 90 vi Chapter 2 Numerical Methods............................ 96 2.1 The Finite Element Method.................... 97 2.1.1 The Rayleigh-Ritz Method................ 98 2.1.2 The Galërkin Method................... 102 2.2 Details of the Finite Element Method.............. 103 2.2.1 The Basis........................... 104 2.2.2 Shape Functions...................... 106 2.2.3 Computation of the Stiffness Matrix........... 109 2.3 Adding Time Dependence..................... 110 2.4 Numerical Methods for Evolutionary Equations........ 112 2.4.1 Euler Methods....................... 113 2.4.2 Other Methods....................... 117 2.5 Error Estimates for the Finite Element Method......... 120 2.6 Discretization of Differential Forms............... 126 2.6.1 Approximation in Hilbert Complexes.......... 128 2.6.2 Approximation with Variational Crimes........ 130 2.6.3 Polynomial Spaces and Error Estimates for Differen- tial Forms.......................... 133 Chapter 3 Some Methods for Nonlinear Equations................ 141 3.1 Overview............................... 142 3.2 Linearizing the Equation...................... 154 3.3 Adding Time Dependence..................... 156 3.4 Newton’s Method.......................... 157 3.4.1 Kantorovitch’s Theorem.................. 160 3.4.2 Globalizing Newton’s Method.............. 161 II Applications to Evolution Problems 164 Chapter 4 Approximation of Parabolic Equations in Hilbert Complexes... 165 4.0 Abstract................................ 165 4.1 Introduction............................. 166 4.2 The Finite Element Exterior Calculus.............. 172 4.2.1 Hilbert Complexes..................... 172 4.2.2 Approximation of Hilbert Complexes.......... 177 4.2.3 Extension of Elliptic Error Estimates for a Nonzero Harmonic Part....................... 182 4.3 Abstract Evolution Problems................... 194 4.3.1 Overview of Bochner Spaces and Abstract Evolution Problems.......................... 195 4.3.2 Recasting the Problem as an Abstract Evolution Equa- tion.............................. 198 4.4 A Priori Error Estimates for the Abstract Parabolic Problem. 201 vii 4.5 Parabolic Equations on Compact Riemannian Manifolds... 213 4.6 Numerical Experiments and Implementation Notes...... 228 4.7 Conclusion and Future Directions................ 230 4.8 Acknowledgements......................... 232 Chapter 5 Finite Element Methods for Ricci Flow on Surfaces......... 233 5.1 Introduction............................. 233 5.2 Notation and Conventions..................... 234 5.3 The Ricci Flow on Surfaces..................... 238 5.4 Weak Form of the Equation.................... 242 5.5 Numerical Method......................... 245 5.6 A Numerical Experiment...................... 249 5.7 Conclusion and Future Work................... 252 5.8 Acknowledgements......................... 256 III Appendices 257 Appendix A Canonical Geometries.......................... 258 A.1 Introduction to Spectral Geometry................ 258 A.2 Solving Poisson’s Equation..................... 261 A.3 Finding Dirichlet Green’s Functions............... 263 A.4 The Dirichlet Problem....................... 265 A.5 The Neumann Problem...................... 268 Appendix B Examples of Green’s Functions and Robin Masses.......... 272 B.1 In One Dimension.......................... 272 B.2 Two-Dimensional Examples.................... 280 B.3 Two-Dimensional Example: The Hyperbolic Disk....... 292 B.4 Derivations for Neumann Boundary Conditions........ 298 B.5 The Finite Cylinder......................... 303 B.6 Domains with Holes in the Plane and the Bergman Metric.. 308 B.7 Conclusion and Future Work................... 321 Bibliography......................................... 322 Index............................................. 330 viii LIST OF FIGURES Figure 1.1: Vectors forming a parallelepiped..................... 15 Figure 1.2: A nonorientable manifold: the Möbius strip and transition charts; the left and right edge are identified in opposite directions as indi- cated by the black arrows. The interior of the charts are indicated with the respective colored arrows and dashed curve boundaries.. 22 Figure 1.3: Demonstration of the cone condition and its violation: (1.3a): The cone condition. Note that the nontrivial cone fits in the corners (and of course, everywhere else) nicely, although it occasionally requires a rigid motion. (1.3b):...................... 31 Figure 1.4: A 1-form ! (thin black level sets) whose hodge dual ?! (gray field lines) has vanishing trace on the boundary @U. This says the field lines of ?! are tangent to @U........................ 51 Figure 1.5: A form and pseudoform in R2 dual to each other, with the two kinds of boundary conditions in the annulus A {a r b}. (1.5a): dθ, a Æ Ç Ç harmonic form whose Hodge dual has vanishing trace on @A. (“dθ” actually is a form determined by overlaps, θ ( ¼,¼) and...... 54 2 ¡ Figure 1.6: Example of harmonic form on closed manifold (here, a torus).... 55 Figure 1.7: Two generators for the harmonic forms for H˚ 1(A) and H1(A), where A is the annulus {a r b} R2, reflecting the different kinds of Ç Ç ⊆ boundary conditions. Note how different they are, but at the same time, how they are dual in some sense, one having level sets that. 81 1 Figure 2.1: Example tent function constructed for the node 2 ; where the nodes in the mesh are are k , k 0,...,6..................... 104 4 Æ Figure 2.2: The heat equation on a piecewise linear approximation of a sphere (3545 triangles). The solution is graphed in the normal direction of the sphere. The spatial discretization uses a surface finite element method detailed in Chapter4 (based on [28]), and implemented. 117 Figure 2.3: The wave equation on a piecewise linear approximation of a sphere (3545 triangles). The solution is graphed in the normal direction of the sphere. The spatial discretization uses a surface finite element method detailed in Chapter4 (based on [28]), and implemented. 121 Figure 3.1: Graphical illustration of for Newton’s Method on a function f (the graph y f (x) is in blue). At each x on the x-axis, draw a vertical Æ i line (dashed red in the above) to the point (xi , f (xi )). From that point, draw a tangent line (in red). Then xi 1 is the intersection.. 158 Å ix Figure 4.1: A curve M with a triangulation (blue polygonal curve Mh) within a tubular neighborhood U of M. Some normal vectors º are drawn, in red; the distance function ± is measured along this normal. The intersection x of the normal with Mh defines a mapping a from x . 216 Figure 4.2: Approximation of a quarter unit circle (black) with a segment (blue) and quadratic Lagrange interpolation for the normal projection (red). Even though the underlying triangulation is the same (and thus also the mesh size), notice how much better the quadratic.. 217 Figure 4.3: Hodge heat equation for k 2 in a square modeled as a 100