Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions

Noel J. Walkington
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA 15213
noelw@cmu.edu

Abstract

We consider linear first order scalar equations of the form $\rho_t + div(\rho v) + a \rho = f$ with appropriate initial and boundary conditions. It is shown that approximate solutions computed using the Galerkin method will converge in $L^2[0,T;L^2((\Omega)]$ 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 may not have bounded variation. pGet the paper in its entirety as