In this talk I will present the idea that by providing a "natural representation" of the solution (e.g. is the solution is C^k, or does the solution have known jumps, does it have a characteristic structure...), one may devise discretizations that are in principle of arbitrary order, and optimally compact. In many cases, the structure required may be hidden, and thus requires a closer look at the geometrical properties of the underlying operator. I will illustrate these concepts by examining 3 popular PDEs: the linear advection, Poisson's equations with jump discontinuities, and the heat equation.