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Bipartite Graphs and Whitney Form Discretization of Helicity Functionals

P. Robert Kotiuga

Boston Universty

Abstract
Explicit Whitney-form finite element discretizations of helicity functionals have been in the engineering literature for over two decades. These discretizations exhibit a version of the metric invariance enjoyed by the continuum functional and have applications to geometric inverse problems such as impedance tomography and the synthesis of force-free magnetic fields. This talk focuses on simplicial meshes arising from locally logically rectangular meshes and reveals some surprising ties between the spectrum of the operator and topological invariants of the mesh. In particular, it reveals spectral quantities which are topological obstructions to having a globally logically rectangular mesh. These topological invariants, related to spectral asymmetry of the discretized helicity functional, have no obvious connection to the discretized Laplace-Beltrami operator. The techniques used in this presentation are ultimately related to those which arise in the analysis of the graph Laplacian and adjacency matrix of a graph. The subtlety is that the underlying elimination graph in question has nodes which are associated with the edges of the underlying tetrahedral finite element mesh, and each element in the mesh gives rise to three edges in the elimination graph. The analysis is more subtle than usual in that the elimination graph is oriented, so that nonzero entries of the adjacency matrix acquire signs. This talk will shy away from the general theory. It will construct logically rectangular meshes for which the elimination graph falls into three disconnected components. For any fixed geometric model, the topological obstructions to having a logically rectangular mesh developed in this talk are then tied to obstructions to having an elimination graph which consists of three separate connected components.

Speaker’s Bio

Prof. Kotiuga received his B.Eng., M. Eng., and Ph.D. from McGill University in 1981, 1982, and 1985 respectively. After a post-doc at MIT, he joined Boston University’s Department of Electrical and Computer Engineering in 1987. Over the years he has held visiting appointments at MIT (Cambridge MA), ETH (Zurich), U. Pau (France), and TUT (Finland). His research focuses on topological aspects of 3-dimensional problems in computational electromagnetics. Two recent books are:

  • Gross, P.W., Kotiuga, P.R., /Electromagnetic Theory and Computation: A Topological Approach./ Cambridge U. Press, 2004, hardback; reissued in 2011, and there’s an e-book version.
  • Kotiuga, P. R., (editor), /A Celebration of the Mathematical Legacy of Raoul Bott, /*CRM Proceedings & Lecture Notes,** *Vol.: 50, Amer. Math. Soc., 2010.