Adaptive high-resolution finite element schemes for transport problems
are presented. A new goal-oriented error estimator is obtained
building on the duality argument. The error in the quantity of
interest is expressed in terms of a linear target functional. The
Galerkin orthogonality error caused by flux limiting is taken into
account and provides a useful criterion for mesh adaptation. Gradient
averaging is invoked to separate the element residual and diffusive
flux errors without introducing jump terms. A decomposition of global
errors into nodal (rather than element) contributions is shown to be
essential. Practical aspects of mesh refinement and coarsening are
discussed. Numerical results and adaptive meshes are presented for
steady hyperbolic and elliptic problems in two dimensions.