It is well known that discrete solutions to the convection-diffusion
equation contain nonphysical oscillations when boundary layers are present
but not resolved by the discretisation. For the Galerkin finite element
method with linear elements on a uniform 1D grid, a precise statement as
to exactly when such oscillations occur can be made, namely, that for a
problem with mesh size h, constant advective velocity and different values
at the left and right boundaries, oscillations will occur if the mesh
P\'{e}clet number $P_e$ is greater than one. In 2D, however, the situation
is not so well understood. In this talk, we present an analysis of a 2D
model problem on a square domain with grid-aligned flow which enables us
to clarify precisely when oscillations occur, and what can be done to
prevent them. We prove the somewhat surprising result that there are
oscillations in the 2D problem even when $P_e$ is less than 1. Also, we show that there
are distinct effects arising from differences in the top and bottom
boundary conditions (equivalent to those seen in 1D), and the non-zero
boundaries parallel to the flow direction.