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In this paper we treat three problems on a two-dimensional `puncturedperiodic 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 boundaryconditions on the boundary of the box $\Omega=(-L,L)^2$, and Dirichlet boundaryconditions on the circumference of the disc. In this setting we consider thePoisson equation, the Stokes equations, and the time-dependent Navier-Stokesequations, all with a fixed forcing function $f$ (which must satisfy$\int_\Omega f=0$ for the stationary problems), and examine the behaviour ofsolutions as $r\to0$. In all three cases we show convergence of the solutionsto those of the limiting problem, i.e.\ the problem posed on all of $\Omega$with periodic boundary conditions.

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