Numerical calculation of localized eigenfunctions of the Laplacian

It is well known that for some planar domains, the Laplacian eigenfunctions are localized in a small region of the domain and decay rapidly outside this region. We address a shape optimization problem of minimizing or maximizing the $L^2$ norm of the eigenfunctions in some sub-domains. This problem is solved by a numerical method involving the Method of Fundamental Solutions and the adjoint method for a fast calculation of the shape gradient.

Several numerical simulations illustrate the good performance of the method.