## Primal-Dual Weak Galerkin Finite Element Methods for PDEs

- Feb. 1, 2019
- 2:30 p.m.
- LeConte 317R

## Abstract

Weak Galerkin (WG) finite element method is a numerical technique for PDEs where the differential operators in the variational form are reconstructed/approximated 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 methods and other existing finite element 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 methods are well suited to a wide class of PDEs by providing the needed stability and accuracy in approximations. The speaker will present a recent development of WG, called "Primal-Dual Weak Galerkin (PD-WG)", for problems for which the usual numerical methods are difficult to apply. 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 method 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. An abstract framework for PD-WG will be presented and discussed for its potential in other scientific applications.