hp -Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems

We consider the {a posteriori} error analysis of hp-discontinuous Galerkin finite element approximations to first-order hyperbolic problems. In particular, we discuss the question of error estimation for linear functionals, such as the outflow flux and the local average of the solution. Based on our {a posteriori} error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local polynomial-degree variation and local mesh subdivision. The theoretical results are illustrated by a series of numerical experiments.