On nonlinear artificial viscosity, discrete maximum principle and hyperbolic conservation laws

A finite element method for Burgers equation is studied. The method is analyzed using techniques from stabilized finite element methods and convergence to entropy solutions is proven under certain hypotheses on the artificial viscosity. In particular we assume that a discrete maximum principle holds. We then construct a nonlinear artificial viscosity that satisfies the assumptions required for convergence and that can be tuned to minimize artificial viscosity away from local extrema. The theoretical results are exemplified on a numerical example.