  A posteriori error estimates are an important tool to bound discretization
errors in terms of computable quantities avoiding regularity conditions that
are often difficult to establish. For non-linear and non-differentiable
problems, problems involving jumping coefficients, and finite element methods
using anisotropic triangulations, such estimates often involve large factors,
leading to sub-optimal error estimates. By making use of convex duality
arguments, exact and explicit error representations are derived that avoid such
effects.
