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.