hp-Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems: Error Analysis and Adaptivity

Abstract

We develop the a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for linear and nonlinear hyperbolic problems. Sharp a posteriori error bounds are derived using a duality argument. The 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. We implement the error bounds into an hp-adaptive finite element algorithm.