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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 $h$).

enter image description here

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  • $\begingroup$ Is the upper bound of your domain always a graph? I.e., is the function you show always single-valued? $\endgroup$ – Wolfgang Bangerth Apr 15 '15 at 11:38
  • $\begingroup$ @WolfgangBangerth No, it can be multivalued $\endgroup$ – Nick P Apr 15 '15 at 15:37
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    $\begingroup$ Too bad. I was hoping that you could just do a vertical stretching to transform it all into a rectangular domain. $\endgroup$ – Wolfgang Bangerth Apr 15 '15 at 18:49
  • $\begingroup$ @WolfgangBangerth Indeed. I have another prescription of $X$ that is in a conformal frame resulting in a rectilinear domain, and hence the results are trivial to extend to depth. Unfortunately, this method is computationally way more expensive to time step, hence satisfying conservation of total effort. $\endgroup$ – Nick P Apr 15 '15 at 19:17
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You may wish to look into boundary integral equation (BIE) methods. These reformulate homogeneous (zero forcing) PDEs into integral equations on the boundary of the geometry, thereby reducing a $d$-dimensional problem to a $d-1$ dimensional problem. These are often the most flexible methods for dealing with complex and moving geometries.

In your case, you would only have to solve a 1D problem for a density (which is then used to compute your solution in the interior) along your geometry boundary. Systems arising in these methods are smaller and well-conditioned (usually) but are typically fully dense. However, there are recent developments which also make numerical discretization of such problems more tractable (notably the Fast Multipole method of Greengard/Rokhlin, advanced quadrature rules, and work generalizing these ideas to the H-matrix and algebraic setting by Martinsson, Gillman, Bremer, and some others).

Alternatives are to use volume-based methods like finite elements and mesh the geometry using something like Delaunay triangulation or a more robust mesh generator (which wouldn't be too difficult for your geometry). You could also look into the cut-cell/immersed boundary method, where you use a structured grid to represent the solution, and take a complicated boundary into account only in the cells which it intersects.

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Since your equation is time-independent, what you seem to have are a sequence of decoupled problems (one for each instant in time). If I am wrong and there is some coupling between them, you should clarify your question.

Any method for solving parabolic problems on arbitrary domains will work fine. The most well-known are standard finite element methods.

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If you are interested in numerical (approximative) solution, one of many methods that I would recommend (and I use it) is an "immersed interface" method (or several other similar methods with similar names like ghost fluid methods etc). In such a case you need to specify boundary conditions on your (moving and fixed parts of) boundary.

The idea is that your moving boundary is represented as a zero set of some function (the level set function, usually a signed distance function). To solve the Laplace equation numerically, you can use very simple finite difference method on structured (quadrangle) grid that involves always your moving boundary. As you want to solve it only for grid nodes where your level set function is (e.g.) negative and not there where it is positive, you have to apply some "extension" (extrapolation) for the nodes next to your boundary. If your boundary conditions on the moving boundary are Dirichlet type (the values of solution are prescribed) then this extension is straightforward.

As I am not sure if you are interested in numerical solution, I stop my description here. As a literature I would recommend:

F. Gibou, R.P. Fedkiw, L.T. Cheng, M. Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys. 176 (2002) 205-227

that is available on the net and you should look only for the stuff dealing with Poisson equation.

I was using such method for moving groundwater table:

P. Frolkovič: Application of level set method for groundwater flow with moving boundary. Advances in Water Resources, Volume 47, October 2012, Pages 56–66

Fancy animation ;-) below

animation of groundwater table

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