Peking University Overseas Distinguished Scholar Lectures--Finite Element Exterior Calculus and its Applications
Douglas N. Arnold is the McKnight Presidential Professor of Mathematics at the University of Minnesota. Prof. Arnold's research interests include numerical analysis, partial differential equations, mechanics, and in particular, the interplay between these fields. Much of his work concerns the computer solution of partial differential equations, focusing on the development and understanding of methods for simulating physical phenomena ranging from the deformation of elastic plates and shells to the collision of black holes. Around 2002 he initiated the finite element exterior calculus, a new approach to the stability of finite element methods based on geometric and topological structure underlying the relevant partial differential equations.
Several of his journal articles are among the most cited in mathematics according to MathSciNet and he was designated a highly cited research by Thomson Reuters both times they released such a list, in 2001 and 2014. In 1991 he was awarded the first International Giovanni Sacchi Landriani Prize. He is highly sought after as a speaker and has delivered plenary lectures at the International Congress of Mathematicians in 2002 and the Joint Mathematics Meetings in 2009. From 2001 through 2008, Prof. Arnold served as director of the Institute for Mathematics and its Applications (IMA). In 2008 he was awarded a Guggenheim Fellowship, and, in 2009, was elected a foreign member of the Norwegian Academy of Science and Letters. In 2009 and 2010, Arnold served as President of the Society for Industrial and Applied Mathematics(SIAM). In 2011 he was elected a Fellow of the American Association for the Advancement of Science (AAAS), and in 2012 he was appointed a Fellow of the American Mathematical Society (AMS). In 2013 he was the recipient of the SIAM Prize for Distinguished Service to the Profession. Homepage: http://www.ima.umn.edu/~arnold
Titles and Abstracts
Finite element exterior calculus (FEEC) is a transformative approach to designing and understanding numerical methods for partial differential equations. FEEC was first explicitly presented in Beijing at the International Congress of Mathematicians in 2002. In 2006 the first major paper appeared, introducing the phrase "finite element exterior calculus" and focusing on simplicial finite element discretizations of the de Rham complex. The theory was extended significantly with a second paper in 2010 in which the central role of Hilbert complexes was highlighted. These two papers are both among the 35 most highly cited papers in mathematics published in their respective years. Since then hundreds of papers and dozens of theses have been written on FEEC and its applications, with applications to electromagnetics, plasma physics, incompressible flow, advection, elasticity, complex materials, plate theory, and general relativity. In this lecture series we will discuss both the abstract framework of FEEC and its specific applications to the de Rham complex and to other complexes arising in different applications.
1) Hilbert Complexes and Applications
Hilbert complexes, which bring together chain complexes of homological algebra with unbounded operators on Hilbert spaces, are a fundamental structure in FEEC. We will review the basic ideas of Hilbert complexes, focusing on the simplest case, consisting of a sequence of three Hilbert spaces connected by two closed unbounded operators. Every such complex gives rise to a sort of Laplace operator, as well as associated wave and heat operators. Key concepts to be discussed include duality, harmonic forms, the Hodge decomposition, and Poincare's inequality. In this lecture we will establish the equivalence of several different formulations of the Hodge Laplace problem, and use the mixed weak formulation to prove its well-posedness up to harmonic forms. The theory will be illustrated through the most canonical example of the de Rham complex and its connection to problems such as Maxwell's equation.
2) Discretization of Hilbert Complexes
An essential aspect of FEEC is that we consider the discretization not only of a single operator equation, but of the underlying Hilbert complex. In this lecture we will describe the abstract theory and again illustrate it with applications tied to the de Rham complex. The key features of a successful discretization are two properties: the subcomplex property and the cochain projection property. We shall show how these two hypotheses lead to many crucial consequences, and ultimately establish the stability and convergence of Galerkin discretizations.
3) Finite Element Differential Forms
In this lecture we will review the theory of differential forms and exterior calculus and use it to determine finite element spaces which satisfy the subcomplex and cochain projection properties for the de Rham complex. A crucial role will be played by the Koszul differential and its connection to the de Rham complex through the homotopy formula. Using it we find that there are two basic families of finite element differential forms, both of which generalize the Lagrange finite element spaces in the special case of 0-forms. In this way we unify, through a single construction, a great variety of mixed finite element methods that had been constructed over preceding decades. Elements from these two families can be combined to give four families of stable mixed finite element methods for the Hodge Laplacian.
4) The Periodic Table of Finite Elements
Besides the two families of simplicial finite element differential forms discussed above, there are two fundamental families of finite element differential forms on cubical meshes. One family is based on a tensor-product construction and recaptures most of known mixed finite elements on cubes, while the other family is largely newly discovered, guided by FEEC. Again the Koszul differential and the homotopy formula play an essential role. Together the four families form a sort of periodic table of finite element differential forms. Key issues, such as unisolvence, can be handled in a uniform fashion. With the cubical elements, issues arise of approximation properties when the elements are deformed by nonaffine multilinear maps.
5) Other Complexes
One of the most important directions for future development of FEEC is its application to complexes other than the de Rham complex. One success in this area is the development of stable mixed finite elements for elasticity, a 50-year old challenge that has been radically advanced and essentially resolved based on FEEC. In that case, as well as others, a key component has been the derivation of new Hilbert complexes by combining old ones. Other applications to be discussed include a FEEC approach to the biharmonic plate equation and a new approach to the Einstein field equations of general relativity.
Lecture Hall, Jiayibing Building, Jingchunyuan 82, BICMR, Peking University
Lecture 1: 16:00pm-18:00pm, August 15, 2015
Lecture 2: 16:00pm-18:00pm, August 16, 2015
Lecture 3: 9:00am-11:00am, August 17, 2015
Lecture 4: 14:00pm-16:00pm, August 17, 2015
Lecture 5: 9:00am-11:00am, August 18, 2015
Jun Hu (Peking University, China)
Jinchao Xu ( Pen State University, USA, & Peking University, China)
Office of International Relations
School of Mathematical Sciences, Peking University
Beijing International Center for Mathematical Research
Contact: Miss He Liu