Solutions to certain partial differential equations exhibit strongly anisotropic features concentrated on lower dimensional manifolds. For instance, solutions to transport dominated problems exhibit shocks and singularly perturbed convection diffusion reaction equations with dominating convection or dominating reaction. In this talk, we first present a novel anisotropic adaptive refinement scheme for transport dominated problems and show its optimally sparse approximation properties for piecewise smooth functions governed by anisotropic features. And then we compare our adaptive scheme with anisotropic triangulation scheme to discuss how each of schemes preforms in order to effectively encode anisotropic features. Finally, we will show some numerical examples for first order transport equations.