Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain

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.