# Advanced Numerical Methods in the Mathematical Sciences

## Junping Wang, National Science Foundation Weak Galerkin Finite Element Methods for $$\text{div}$$-$$\text{curl}$$ Systems

• Junping Wang
• Chunmei Wang

### Abstract

This talk shall introduce a new numerical technique, called the weak Galerkin finite element method (WG), for partial differential equations. The presentation will start with the second-order elliptic equation, for which WG shall be applied and explained in detail. 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 $$\text{curl}$$, etc. These weak differential operators shall serve as building blocks for WG finite element methods for other classes of partial differential equations, such as the Stokes equation, the biharmonic equation for thin plate bending, the Maxwell equations in electron magnetics theory, and the $$\text{div}$$-$$\text{curl}$$ systems. In particular, the speaker will demonstrate how WG can be applied to the $$\text{div}$$-$$\text{curl}$$ systems. A mathematical convergence theory shall be briefly discussed for this application. The talk should be accessible to graduate students with adequate training in computational methods.