10月21日 王军平教授学术报告

发布时间:2016-10-19

题 目：Weak Galerkin Finite Element Methods

报告人：Dr. Junping Wang, Program Director, National Science Foundation

时 间： 10月21日（周五） 上午8：30-9：30

地 点：南校区新数学楼416

报告摘要：
Weak Galerkin (WG) is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc.) in the weak forms are approximated by discrete 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 weak forms for the underlying PDEs. Weak Galerkin is a natural extension of the classical Galerkin finite element method with advantages 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 approximation.
This talk will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of discrete weak gradient (or discrete weak derivative) will be introduced and discussed for its role in the design of weak Galerkin finite element schemes. The speaker will then introduce a general notion of weak differential operators, such as weak Hessian, weak divergence, and weak curl etc. These weak differential operators shall serve as building blocks for WG finite element methods for other class of partial differential equations, such as the Stokes equation, the biharmonic equation, the Maxwell equations in electron magnetics theory, div-curl systems, and PDEs in non-divergence form.
Following the introduction of basic principles of WG, The speaker will introduce a newly developed finite element technique, called Primal-Dual WG, for some model problems where the trial and test spaces are different. The primal-dual method enhances the primal (original) equation by its dual with homogeneous data. The two equations are linked together by using appropriately defined stabilizers commonly used in weak Galerkin finite element methods. The primal-dual technique will be discussed for three model problems: (1) second order elliptic equation in nondivergence form, (2) stationary linear convection equations, and (3) elliptic Cauchy problem which is generally ill-posed. The talk should be accessible to graduate students with adequate training in computational methods.

报告人简介：
Dr. Junping Wang received his PhD in Mathematics from the University of Chicago in 1988. Following his PhD, he went to Cornell University as a postdoctoral associate at the Mathematical Sciences Institute. He then joined the faculty of the University of Wyoming at the end of 1989, and served as the Director for the Institute for Scientific Computing since 1997. He was a visiting faculty of Texas A&M University in 1995-1997. In 1999, Dr. Wang joined the faculty of Colorado School of Mines, and served as Assistant Head of the Department of Mathematical and Computer Sciences. In 2002, Dr. Wang served as Acting Head for the same department at Colorado School of Mines. In 2003, Dr. Wang moved to National Science Foundation (NSF) as a Program Director in the Division of Mathematical Sciences. His primary duty at NSF is to promote and support research and training in all fields of Computational Mathematics. At NSF, Dr. Wang has been actively involved in various foundation-wide and cross agency activities of interdisciplinary nature. He is an advocator and supporter of computational methods with innovation.
Dr. Wang’s research interests are primarily in numerical methods for partial differential equations. His main research contribution is in the theoretical aspect of numerical PDEs, such as algorithm design, convergence and error analysis for finite element methods, domain decomposition and multigrid methods. His recent work is focused on the development of a new computational technique, called weak Galerkin (WG) finite element method.