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.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.

In this talk, we start with the mathematical modeling of air-water interaction in the framework of the interface problem between two incompressible inviscid fluids under the influence of gravity/surface tension. This is a nonlinear PDE system involving free boundary. It is generally accepted that wind generates surface waves due to the instability of shear flows in this context. Based on the linearized equations about shear flow solutions, we will discuss the classical Kelvin--Helmholtz instability etc. before we illustrate Miles' critical layer theory.

In this seminar, Prof. John Etnyre will begin this talk by discussing a classical question concerning periodic motions of particles in classical physics. In trying to better understand this question we will develop the notion of a symplectic structure. This is a fundamental geometric object that provides the "right way" to think about classical mechanics, and many many other things too. We will then indicate how modern ideas can be used to give, at least partial, answers to our initial naive questions about periodic motions.