Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part II: Hyperbolic Equations

In this paper we consider stabilized 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 stabilized methods: the Galerkin least squares method, the continuous interior penalty method, and the discontinuous Galerkin method. We consider both the standard stabilization methods and the optimization-based method introduced The main idea of the latter is to write the stabilized method in an optimization framework and select the discrete function for which a certain cost functional, in our case the stabilization term, is minimized. Some numerical examples illustrate the theoretical investigations. 1. Introduction. Several finite element methods have been proposed for the computation of hyperbolic problems, such as the SUPG method [5, 16], the discon-tinuous Galerkin (DG) method [19, 18, 17], and several different weakly consistent, symmetric stabilization methods for continuous approximation spaces [14, 11, 9, 3]. In most of these cases, however, the analysis relies on the satisfaction of a coercivity condition. Indeed if a scalar hyperbolic transport equation