Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions

We consider linear first order scalar equations of the form ρt + div(ρv) + aρ = f with appropriate initial and boundary conditions. It is shown that approximate solutions computed using the discontinuous Galerkin method will converge in L[0, T ;L(Ω)] when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions. In particular, v need not be Lipschitz, so characteristics of the equation may not be defined, and the solutions being approximated my not have bounded variation.