I have really enjoyed performing the method of separation of variables to identify the eigenfunctions and nodal lines (the set of points for which each eigenfunctions vanishes) of the 2-D wave equation $u_{tt}=c^2 \Delta u$ over circular and rectangular domains.
However, I really want to see what the nodal lines look like on other domains. Specifically, given a domain $D\subset \mathbb{R}^2$, I wish to approximate the nodal lines for the Dirichlet eigenvalue problem:
$\left\{\begin{matrix} \Delta u=\lambda u & (x,y)\in D\\ u=0 & (x,y)\in \partial D \end{matrix}\right.$
I know that this is impossible for a general domain, so I first want to find a way of approximating these lines for a rectangle and a polygon (let's say a pentagon). I know we can obtain an analytical solution for the rectangle but the same cannot be said for other domains, which is why I seek a numerical technique.
The problem is that I really don't know where to start. I don't know whether I first need to approximate each eigenfunction before finding its zeros, or whether there is a way of identifying the nodal lines without knowing the eigenfunctions - perhaps by a symmetry argument.
There are many numerical methods out there but I don't know what would be best to solve this problem. Unfortunately, I have had no experience with numerical techniques and I fear that the level of mathematics is far beyond my capabilities.
However, I would really appreciate some advice on how to tackle this problem; I know it's not easy, but I would honestly appreciate any advice I can get before I continue my quest. If somebody could point me in the direction of an algorithm/method which will do the job, then that would be perfect.
Finally, I would like to apologise for the lack of clarity in my question. I am new to writing mathematics and I often find it difficult to convey my ideas in written form. Thus, if you do have any questions then please ask away.
I uploaded Matlab example to Github. It requires PDE Toolbox and is quite rude but should work for any 2D domains that are supported by pdetool's initmesh function. I just plotted contours at solution equals zero to approximate nodal lines. You should create a better function for that job I presume.