I solved an eigenvalue problem on a domain D= D1 U D2 using the matrix construction as explained in the Freefem documentation. Each domain is triangulated and the meshes are adapted with the corresponding metric. My operators are symmetric and I know that theoretically the zero eigenspace is three-dimensional.
My operator does not have the same expression in each domain. And even if my Freefem code was computed, numerically I do not have a triple zero eigenvalue. I think this is because the basis is not the same for D1 and D2, and it is not taken into account in the assembling of matrices. I did not have any warning in my code because I was working with several domains.
We found one method to solve this problem with C++, and we think that maybe it can be added on Freefem software.
I can send more details in a PDF file if you are interested.