LECTURE # 1: DISCONTINUOUS GALERKIN METHODS: MOTIVATION AND THEIR ORIGIN RAYTCHO D. LAZAROV AND SATYENDRA TOMAR A . In this lecture we introduce the main ideas of the discontinuous Galerkin approximations that are based on weak formulation of differential equation and the boundary conditions. First we introduce some notations and then we give a very simple example of one dimensional transport problem, which motivates and serves as a background for the approximation to the multidimensional transport problem. We discuss two equivalent weak formulations for the multidimensional problem. We conclude with some simple error estimates. 1. P The first variant of DG finite element approximation was considered by Reed and Hill in 1973 [4] who considered the transport equation for the neutron flux ux, Ψ , Ψ S 3 , S 3 the unit sphere in R d , Ω R d , d 2, 3 a bounded domain with a boundary Ω: Ψ ∇u σu S u f in Ω, (1.1a) Ψ n u g 0 on Γ inc x Ω : n Ψ 0 . (1.1b) We denote by ∇u a gradient of the scalar function u, by ∇ b a divergence of a vector-field b, by σ x such that σ x σ 0 0 the cross section, by n the external unit normal vector, and by S u a certain integral operator on S 3 . Here Γ inc and Γ out ΩΓ inc denote the inflow and outflow boundary, respectively. After splitting from Su the isotropic part of the solution and leaving the rest on the previous iteration (a procedure is often called source iteration) one gets a transport equation. Γ Γ inc out F 1. Neutron transport This problem is a particular case of more general class of first order hyperbolic equations which are formally obtained from (1.1) by replacing Ψ by space dependent divergence-free vector-field b x , i.e. ∇ b divb 0. Thus, we get the following problem, numerical treatment of which is Date: Draft version: October 31, 2005. 1991 Mathematics Subject Classification. 65N30, 65N15. 1