Colloquium

Continuous and discrete maximum principles are formulated for scalar transport equations. Relevant a priori bounds are enforced within the framework of algebraic flux correction. A family of nonlinear high-resolution finite element schemes is presented and combined with adaptive mesh refinement. The derivation of computable error indicators is discussed in some detail. Gradient recovery techniques are revisited and integrated into a new goal-oriented a posteriori error estimate. The methodology to be presented builds on the duality argument and features a node-based approach to localization of errors in the quantity of interest, as represented by a linear target functional. A possible violation of Galerkin orthogonality is taken into account in a simple and natural way. The use of an averaged gradient makes it possible to obtain a nonoscillatory distribution of weighted residuals without introducing jump terms. The weights are determined using the difference between the linear and quadratic finite element interpolants of the dual solution. The benefits of mesh adaptation are illustrated by numerical results for scalar conservation laws and hyperbolic systems.