This talk will introduce a primal-dual finite element method for variational problems where the trial and test spaces are different. The essential idea behind the primal-dual method is to formulate the original problem as a constrained minimization problem. The corresponding Euler-Lagrange formulation then involves the primal (original) equation and its dual with homogeneous data. The two equations are linked together by using properly-defined stabilizers commonly used in weak Galerkin finite element methods. The primal-dual method will be discussed for three type of model problems: (1) second order elliptic equation in nondivergence form, (2) steady-state linear convection equations, and (3) elliptic Cauchy problems.

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. The goal of this talk is to demonstrate some of these advantages in numerical PDEs.

The talk will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of weak gradient 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. The speaker will then discuss the primal-dual technique for the three model problems. The talk should be accessible to graduate students with adequate training in computational mathematics.