Abstract
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 in [E. Burman, SIAM J. Sci. Comput., 35 (2013), pp. A2752--A2780]. 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.
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