I have the following setting:An irregularly-shaped domain, expressed as a mesh of pointsA Laplacian operator, together with boundary conditionsI am looking for the eigenvalues of that operator, i.e. functions (expressed on the mesh, though clearly approximating an underlying continuous function) satisfying the boundary conditions and$$\triangledown^2 f = \lambda f$$for some function $f$ and constant $\lambda$. I don't know $\lambda$ a priori, in fact part of the problem is to find the possible values!What I got so far for the special case, where the boundary conditions are a constant (e.g. 0s):The Laplacian operator, when acting on a discrete mesh of points, becomes just a linear operation - e.g. a matrix transformationSo I can just look for the eigenvalues of thatThat works pretty well, but doesn't generalise so well, I think. E.g. now I would like to solve the problem with boundary conditions such that on the boundary$$\frac{ \partial f }{ \partial n } = 0$$and now I  don't think I can just express that as a matrix eigenvector problem - or maybe I can?For context, I am solving the problem of drum vibrations on an irregular-shaped domain. Given a domain, the solutions to the problem above give the shape of vibrations of the drum, along with the associated frequencies (which can be deduced from the eigenvalues). Here the boundary conditions is $f=0$ on boundary, and it solves nicely - I can produce a pretty picture like this, for example (here my domain is a semi-circle):However, now I would like to extend this to water waves in a cyllindroid container. The boundary condition becomes that of partial derivatives vanishing at the boundary described above - and I feel out of ideas. All help will be highly appreciated.
Eigenvalues of a Laplacian operator on an irregular mesh
pde;eigenvalues;eigensystem