Date & Time: Wednesday 10th August , 2011 11:30-12:30.

Venue: Ramanujan Hall, Mathematics Department

Title: Dual Finite Elements and Discrete Hodge Stars

Speaker:Prof. Chandrajit Bajaj, University of Texas at Austin, USA

Abstract: Real polynomial scalar- or vector-valued interpolants (finite elements) are widely used to determine the numerical solution of Partial Differential Equations (PDEs).

While 'primal' finite element methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of usually a simplicial mesh, a 'dual' finite element associates DoFs with the geometric duals of the primal mesh . We analyze the approximation of solutions to PDEs over domains discretized by the choice of both primal and dual finite elements. These finite elements are said to be "conforming" and "stable" if the sequence of discrete solution spaces they span have the same cohomology as the de Rham complex of the domain. Using the language of discrete exterior calculus, I will unify prior approaches as well as motivate the discretization and numerical stability of primal and dual finite elements. The choice of the discrete Hodge star mapping between primal and dual finite element spaces, also plays a crucial role in showing that primal-dual methods can attain the same approximation power with regard to discretization stability as only primal methods and in some circumstances, offer improved numerical stability properties.

This is joint work with Andrew Gillette.

About the Speaker:Chandrajit Bajaj is the director of the Center for Computational Visualization, in the Institute for Computational and Engineering Sciences (ICES) and a Professor of Computer Sciences at the University of Texas at Austin. Bajaj holds the Computational Applied Mathematics Chair in Visualization. He is also an affiliate faculty member of Mathematics, Electrical Engineering, Bio-Medical Engineering, Neurobiology, and the Institute of Cell and Molecular Biology. He additionally participates actively in the activities of the Center for Learning and Memory, and the Center for Perceptual Systems. He graduated from the Indian Institute of Technology, Delhi in 1980, with a Bachelor's Degree in Electrical Engineering, and received his M.S. and Ph.D. degrees in Computer Sciences from Cornell University, in 1983, and 1984 respectively. He is on the editorial boards for the International Journal of Computational Geometry and Applications, the ACM Computing Surveys, the SIAM Journal on Imaging Sciences and the Journal of Visualization. He is on numerous national and international conference committees, has served as a scientific consultant to national labs, and national science foundations of Spain, Austria, Korea, and China. He is a fellow of the American Association for the Advancement of Science (AAAS) and a fellow of the Association of Computing Machinery (ACM).