Georgia Institute of Technology College of Sciences

Weak Galerkin (WG) is a new finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation. In this talk, the speaker will introduce a general framework for WG methods by using the second order elliptic problem as an example. Furthermore, the speaker will present WG finite element methods for several model PDEs, including the linear elasticity problem, a fourth order problem arising from fluorescence tomography, and the second order problem in nondivergence form. The talk should be accessible to graduate students with adequate training in computational mathematics.

I will finish my overview of the de Rham fundamental group by reviewing two explicit calculations: Deligne's completely concrete description of the unipotent fundamental group of the projective line minus three points in terms of the free nilpotent Lie algebra on two generators, and Chen's general calculation of the unipotent fundamental group of a manifold in terms of iterated integrals.

In this talk, first, I'll briefly go over my proof of the conjecture that there are only afinite number of hyperbolic 3-manifolds of bounded volume and trace field degree. Then I'lldiscuss some conjectural pictures to quantitative results and applications to other similarproblems.

I present a formalism and an computational scheme to quantify the dynamics of grain boundary migration in polycrystalline materials, applicable to three-dimensional microstructure data obtained from non-destructive coarsening experiments. I will describe a geometric technique of interface tracking using well-established optimization algorithms and demonstrate how, when coupled with very basic physical assumptions, one can effectively measure grain boundary energy density and mobility of a given misorientation type in the two-parameter subspace of boundary inclinations. By doing away with any specific model or parameterization for the energetics, I seek to have my analysis applicable to general anisotropies in energy and mobility. I present results in two proof-of-concept test cases, one first described in closed form by J. von Neumann more than half a century ago, and the other which assumes analytic but anisotropic energy and mobility known in advance.