In this lecture, firstly we will talk about a unified analysis of discontinuous Galerkin methods for elliptic problems. Secondly, we will extend the interior penalty discontinuous Galerkin method to fourth-order elliptic boundary value problems. Thirdly, we will discuss discontinuous Galerkin methods for the linear elasticity problem. In particular, we will prove that the discontinuous Galerkin methods are locking-free and optimal. Fourthly, we will talk about discontinuous Galerkin discretizations for the Stokes problem which preserve the divergence-free condition exactly. Finally, if there is time, we will give an idea on uniformly convergent iterative methods for discontinuous Galerkin discretizations of elliptic problems.
Prerequisites: Basic training in finite element method and numerical method for partial differential equations
Level of Students : Graduate students from any grade
 D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discon- tinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39:17491779, 2002.
 S.C. Brenner and L.Y. Sung, C0 Interior penalty methods for fourth-order elliptic boundary value problems on polygonal domains, Journal of Scientific Computing, 22-23 (2005), 83-118.
 Qingguo Hong, Johannes Kraus, Jinchao Xu, Zikatanov Ludmil, A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations, Numerische Mathematik, 2016, 132, 23-49.
 Ayuso de Dios, B. and Zikatanov, L. Uniformly convergent iterative methods for dis- continuous Galerkin discretizations, Journal of Scientific Computing, 2009,40,4-36.