Weak Galerkin (WG) is a new finite element method for partial differential equations (PDEs) 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 finite element methods (FEM) is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on conventional weak forms for the underlying PDE problems. Weak Galerkin is a natural extension of the classical Galerkin finite element method and has advantages over FEM in many aspects. 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 approximations. In this talk, the speaker will introduce the basic ideas and a general framework for WG methods by using the second order elliptic equation as a motivating example. Furthermore, the speaker will discuss WG finite element methods for two model PDEs arising from applications, including a fourth order problem arising from fluorescence tomography, and the second order elliptic problem in nondivergence form.

Dr. Chunmei Wang is originally from China and received her PhD from Nanjing Normal University, China in June 2014. Chunmei is currently a tenure-track assistant professor in Texas State University. Before joining Texas State University, she was a visiting assistant professor in Georgia Institute of Technology from August 2014 to May 2016. Her research interests fall under the broad heading of numerical methods and scientific computing for problems in science and engineering governed by partial differential equations (PDEs), with the goal to obtain approximate solutions for PDEs arising from physical, engineering, and biological applications through predictive modeling and computational simulation. She has received two NSF awards in 2015 for her work in computational mathematics and applications. With the support from NSF, she is developing an efficient numerical solver for a fourth order PDE problem arising from fluorescence tomography modeling in medical science involving big data. She was one of the organizers of an international conference on "Polytopal Element Methods in Mathematics and Engineering" which was held from October 26-28, 2015 on the campus of Georgia Institute of Technology.