Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problem. Part II: hyperbolic equations

In this paper we consider stabilised finite element methods for hyperbolic transport equations without coercivity. Abstract conditions for the convergence of the methods are introduced and these conditions are shown to hold for three different stabilised methods: the Galerkin least squares method, the continuous interior penalty method and the discontinuous Galerkin method. We consider both the standard stabilisation methods and the optimisation based method introduced in \cite{part1}. The main idea of the latter is to write the stabilised method in an optimisation framework and select the discrete function for which a certain cost functional, in our case stabilisation term, is minimised. Some numerical examples illustrate the theoretical investigations.