18SC002 An Introduction to the high-order upwind Hybridized discontinuous Galerkin methods for partial differential equations

Instructor: Tan Bui-Thanh, ICES, University of Texas at Austin


In this short course, we present a systematic and constructive derivation of hybridized discontinuous Galerkin (HDG) methods using an upwind framework. The key behind our construction is the Godunov method and the Rankine-Hugoniot condition. We will show in details the construction of an upwind HDG framework and its application to constructively derive HDG methods for a large class of partial differential equations (PDEs) including elliptic, parabolic, hyperbolic, and mixed-types PDEs. The participants will learn how to apply this framework to derive HDG methods for convection-diffusion-reactions equations (and its subsets), Maxwell equations, and linearized shallow water equation. The well-posedness of the HDG framework will be discussed and the well-posedness analysis of the HDG formulation for the convection-diffusion-reaction will be carried out in details. The participants will learn a step-by-step convergence analysis of the HDG method for the linearized shallow water equations. We will discuss the extension to nonlinear systems and active research topics.

At the end of the short course, participants will, without wary, be able to construct HDG methods for their applications of interest. They will learn how HDG is implemented in practice and accompanied HDG Matlab codes will be provided at end of the short course.

Outline of the course:

1) Introduction to the Discontinuous Galerkin method for hyperbolic PDEs

2) Derivation of the upwind HDG framework for hyperbolic PDEs

3) Application of the upwind HDG framework to convection-diffusion-reaction equations

4) Application of the upwind HDG framework to Maxwell equations

5) Application of the upwind HDG framework to linearized shallow water equations

6) Well-posedness of the HDG framework and its application to the convection-diffusion-reaction equations

7) A convergence analysis for the upwind HDG method for the linearized shallow equations

8) Extension of the HDG framework to nonlinear PDEs and active research topics

9) Implementation aspects and the details of Matlab codes for convection-diffusion-reaction equations