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Nonlocal Exterior Calculus on Riemannian Manifolds The Pennsylvania State University The Graduate School Department of Mathematics NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS A Dissertation in Mathematics by Thinh Duc Le c 2013 Thinh Duc Le Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 ii The dissertation of Thinh Duc Le was reviewed and approved* by the following: Qiang Du Verne M. Willaman Professor of Mathematics Dissertation Adviser Chair of Committee Long-Qing Chen Distinguished Professor of Material Sciences and Engineering Ping Xu Distinguished Professor of Mathematics Mathieu Stienon Associate Professor of Mathematics Svetlana Katok Director of Graduate Studies, Department of Mathematics *Signatures are on file in the Graduate School. iii Abstract Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries. Variations of these concepts have also been made over the years such as the discrete exterior calculus and the finite element exterior calculus. In this work, motivated by the recent studies of nonlocal vector calculus we develop a nonlocal exterior calculus framework on Riemannian manifolds which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal \interactions" (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Just as in the standard context, we introduce all necessary elements of exterior calculus such as forms, vector fields, exterior derivatives, etc. We point out the relationships between these elements with the known ones in (local) exterior calcu- lus, discrete exterior calculus, etc. We also introduce nonlocal Hodge theory and its connections with existing works. iv Table of Contents Acknowledgments :::::::::::::::::::::::::::::::: vi Chapter 1. Introduction :::::::::::::::::::::::::::: 1 1.1 Local Exterior Calculus and Finite Element Exterior Calculus2 1.2 Discrete Exterior Calculus . .3 1.3 Existing Works in Nonlocal Vector Calculus . .4 1.4 Hodge Theory on Metric Spaces . .5 1.5 Details of this Dissertation . .7 Chapter 2. Nonlocal forms :::::::::::::::::::::::::: 10 2.1 Oriented Simplices and Tuples . 10 2.2 Nonlocal Forms . 11 2.3 Nonlocal Exterior Derivative (D)................ 12 2.4 Codifferential Operator (D∗)................... 17 2.5 NL Laplace-Beltrami Operator (∆) . 21 2.6 NL Hodge Operator (∗)...................... 21 2.7 Nonlocal Wedge Product ^ ).................. 24 nl Chapter 3. Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus ::::::::::::::: 25 3.1 Nonlocal Trivializations (λ)................... 25 3.2 Vector Fields . 27 3.3 NL Sharp (]) and Flat ([) Operators . 27 3.4 NL Vector Calculus . 31 v 3.5 Some special cases of NL vector calculus operators . 37 Chapter 4. Relationships to Local Geometry, Classical Vector Calculus and Discrete Vector Calculus ::::::::::::::::::::: 43 4.1 Relationship between NL geometry and local geometry . 43 4.2 Relationship between NL Vector Calculus and the Local One . 45 4.3 Relationship between NL Laplacian and discrete Laplacian . 46 Chapter 5. NL Hodge Theory ::::::::::::::::::::::::: 50 5.1 Hodge Theory for the Discrete Derivative . 50 5.2 Hodge Theory for the Nonlocal Exterior Derivative . 63 n Chapter 6. Another Model for Nonlocal Exterior Calculus in R ::::: 67 6.1 Nonlocal Forms . 67 6.2 Nonlocal Exterior Derivative (D)................ 68 6.3 Codifferential Operator (D∗)................... 69 6.4 NL Laplace-Beltrami Operator (revisit) . 70 6.5 Hodge Operator (∗)........................ 70 6.6 Sharp (]) and Flat ([) Operators . 73 6.7 Vector Calculus (revisit) . 75 6.8 NL Wedge Product (Λnl).................... 76 Chapter 7. Ongoing and Future Works :::::::::::::::::::: 78 7.1 Ongoing Works . 78 7.2 Future Works . 78 References :::::::::::::::::::::::::::::::::::: 80 vi Acknowledgments I would like to thank Professor Ping Xu and Professor Aissa Wade for giving me the chance to study at Penn State. Without their great support I would not be able to come to Penn State for my Ph.D. I would like to thank my advisor, Professor Qiang Du, for believing and trusting in my capability by giving me the opportunity to be one of his students and investing his research projects and grants in me. He has been always available and willing to help during my studies. I really appreciate that he is very patient with me and gives me great supports on both research and finance. Without his brilliant guidance this dissertation could not be done. I would like to thank Professor Ping Xu and Professor Mathieu Stienon from the Department of Mathematics and Professor Long-Qing Chen from the Depart- ment of Material Sciences and Engineering for helpful discussions and serving on my committee. I also would like to thank Doctor Tadele Mengesha for his help via a lot of discussions with him in my final year at Penn State. I would like to give my appreciations to all the members in my research group for their helps and supports. Finally I would like to thank my family for their great supports especially when I am very far away from my home country. 1 Chapter 1 Introduction Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries [20]. They have made profound influence to the de- velopment of mathematics and they now can be learned from standard text books [33]. Variations of these concepts have also been made over the years. Highly suc- cessful examples include the discrete exterior calculus [10, 17] and the finite element exterior calculus [1] which are extensions to discrete spaces including piecewise lin- ear complexes and finite element functions. They have proved to be useful in the development and analysis of finite element methods. More recently, motivated by the study of fractional diffusion processes and nonlocal electromagnetic media, the fractional exterior calculus has also been developed [9, 21, 34]. In this work, motivated by the recent studies of nonlocal calculus [16, 11], we develop a nonlocal (NL) exterior calculus framework which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal interactions (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Since our work is partly related to local exterior calculus, discrete exterior calculus, existing works on nonlocal vector calculus and a L2 Hodge theory pro- posed by S. Smale et al. First of all we would like to give an literature overview on these theories which are relevant to our work. 2 1.1 Local Exterior Calculus and Finite Element Exterior Calculus The exterior calculus of differential forms, also called Cartan's calculus, is one whose geometric underpinning is the exterior algebra. It was born through a paper in 1899 by Elie´ Cartan ([6]). The first key ingredient of exterior calculus is differential forms. They are an approach to multivariable calculus that is indepen- dent of coordinates. Forms are local in the sense that they are defined pointwise: at any point x in a given manifold M, a p-form ! defines a skew-symmetric p-linear map p ! :(T M) ! x x R and as an operator of x, ! is required to be smooth (here T M is the tangent space x to M at x). Forms can interact with each other via wedge product (exterior algebra). Differential forms provide a unified approach to defining integrals over curves, surfaces, volumes and higher dimensional (Riemannian) manifolds. Another key ingredient is vector fields. These are the dual of 1-forms via musical isomorphisms (sharp and flat operators). Besides these two operators, other key operators include exterior derivative, codifferential, Hodge star, Laplace- Beltrami, etc. We call \intrinsic properties" the relationships between key ingre- dients and key operators which are coordinate-free. Following the same procedure, we define our nonlocal exterior calculus with similar key ingredients and key operators. Our goal is to preserve as many intrinsic properties as possible. However due to nonlocality (almost all operators are integral operators), some intrinsic properties may not be achievable and some operators play less important roles (see Section 1.5 below). 3 Exterior calculus has many applications, especially in geometry, topology, partial differential equations (PDEs) and physics ([2],[13], [32]). Many partial differential equations (PDEs) are related to differential complexes, that is they can be rewritten using exterior calculus in neat forms. For example Maxwell's equations can be written very compactly in geometrized units using forms, exterior derivative and Hodge star. From this point of view, the authors of [1, 2] develop the theory of Finite Element Exterior Calculus, which serves the study on numerical analysis and scien- tific computation of many PDEs. These PDEs are related to differential complexes but they are a fundamental component of problems arising in many mathematical models. This theory is developed to capture the key structures of the L2 de Rham complex and Hodge theory at the discrete level and to relate the discrete and continuous structures, in order to obtain stable finite element discretizations. The authors also develop an abstract Hilbert space framework (Hilbert complex), which captures key elements of Hodge theory
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