Hp‐Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity

Abstract

AbstractIn 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 & Sons, Ltd.