Stabilized hp-Finite Element Methods for First-Order Hyperbolic Problems

We analyze the $hp$-version of the streamline-diffusion (SDFEM) and of the discontinuous Galerkin method (DGFEM) for first--order linear hyperbolic problems. For both methods, we derive new error estimates on quadrilateral meshes which are sharp in the mesh-width $h$ and in the spectral order $p$ of the method, assuming that the stabilization parameter is $O(h/p)$. For piecewise analytic solutions, exponential convergence is established. For the DGFEM we admit very general irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Numerical experiments confirm the theoretical results