Dr. Xiu Ye, University of Arkansas, Little Rock
Dr. Chunmei Wang, Texas Tech University
Weak Galerkin (WG) finite element method is a numerical technique for PDEs where differential operators in the variational form are reconstructed by using a framework that mimics the theory of distributions for piecewise polynomials. The usual regularity of the approximating functions is compensated by carefully-designed stabilizers. The fundamental difference between WG and other existing methods is the use of weak derivatives and weak continuities in the design of numerical schemes based on conventional weak forms for the underlying PDE problems. Due to its great structural flexibility, WG is well suited to a wide class of PDEs by providing the needed stability and accuracy in approximations. There is a recent development of WG, called "Primal-Dual Weak Galerkin (PD-WG)". The essential idea of PD-WG is to interpret the numerical solutions as a constrained minimization of some functionals with constraints that mimic the weak formulation of the PDEs by using weak derivatives. The resulting Euler-Lagrange equation offers a symmetric scheme involving both the primal variable and the dual variable (Lagrange multiplier). PD-WG is applicable to several challenging problems for which existing methods may have difficulty in applying; these problems include the second order elliptic equations in nondivergence form, Fokker-Planck equation and elliptic Cauchy problems. This short course will introduce WG, PD-WG and further discuss the connections between WG and other numerical methods such as HDG.