ABSTRACT
The existence of a maximum principle allows the introduction of comparison functions, which are often used to provide solution bounds and error bounds for approximations. Even for nonlinear problems, the modern theory of viscosity solutions utilises similar techniques. The idea of boundary layers, and the use of asymptotic expansions with some sort of matching, has been the key to understanding viscous flows in two and three dimensions. In two and three dimensions, the interaction between the singularity in the fundamental solution for the second order diffusion operator, and the exponential form of the Green's function for the one-dimensional convection-diffusion operator, is most simply seen by ignoring boundary effects. If one also assumes constant coefficients they can use Fourier analysis to construct the Green's function explicitly.
