  In this paper we treat three problems on a two-dimensional `punctured
periodic domain': we take $\Omega_r=(-L,L)^2\setminus D_r$, where $D_r=B(0,r)$
is the disc of radius $r$ centred at the origin. We impose periodic boundary
conditions on the boundary of the box $\Omega=(-L,L)^2$, and Dirichlet boundary
conditions on the circumference of the disc. In this setting we consider the
Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes
equations, all with a fixed forcing function $f$ (which must satisfy
$\int_\Omega f=0$ for the stationary problems), and examine the behaviour of
solutions as $r\to0$. In all three cases we show convergence of the solutions
to those of the limiting problem, i.e.\ the problem posed on all of $\Omega$
with periodic boundary conditions.
