hp-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity

In this paper we develop the a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for linear and non-linear hyperbolic problems. By employing a duality argument, sharp a posteriori error bounds are derived for certain output functionals of practical interest. These bounds exhibit an exponential rate of convergence under hp-refinement if either the primal or the dual solution is an analytic function over the computational domain. Based on our a posteriori error bounds, we design and implement the corresponding hp-adaptive finite element algorithm to ensure the reliable and efficient control on the error in the prescribed functional to within a user-defined tolerance. Copyright © 2002 John Wiley and Sons, Ltd