Consider a function $X(\xi,\nu)$, $2\pi$ periodic in $\xi$ satisfying $$\nabla^2 X = 0$$in a domain $D$ with $\nabla = (\partial_{\xi},\partial_{\nu})$. If I know the values of $X$ on the boundary $\partial D$, what is the simplest (in a numerical sense) way of finding $X$ in the interior? Here, $\partial D$ is a complex geometry, which will be evolving at each instant of time. It looks like, for example, the graph shown in figure 1 (this is the boundary together with lines at $\xi=0$, $\xi=2\pi$ and $\nu=-h$ for some large constant $h$). 
Solving Laplace's equation on a domain with moving boundary
pde