hp-Version discontinuous Galerkin methods for hyperbolic conservation laws

The development of hp-version discontinuous Galerkin methods for hyperbolic conservation laws is presented in this work. A priori error estimates are derived for a model class of linear hyperbolic conservation laws. These estimates are obtained using a new mesh-dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation. The results generalize and extend previous results on mesh-dependent norms to hp-version discontinuous Galerkin methods. A posteriori error estimates which provide bounds on the actual error are also developed in this work. Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution.