Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws

We consider the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws. In particular, we discuss the question of error estimation for general target functionals of the solution; typical examples include the outflow flux, local average and pointwise value, as well as the lift and drag coefficients of a body immersed in an inviscid fluid. By employing a duality argument, we derive so-called weighted or Type I a posteriori error bounds; these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated. Based on the resulting approximate Type I bound, we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest. The performance of the proposed adaptive strategy and the quality of the approximate Type I a posteriori error bound is illustrated by a series of numerical experiments. In particular, we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ Type II error indicators.