Why solving Helmholtz equation using FEM gives solution when theory says no? I get problem when solving $-\Delta u(x,y) - k^2 u(x, y) = f(x, y)$ in $(0, 1) \times (0, 1)$, $u = 0$ on the boundary. 
When $k^2$ equals an eigenvalue, says $k^2 = 2 \pi^2$, using the Fredholm alternative, theorically we can say the problem is not well-posed, that we cannot obtain a unique solution for the problem. I solve this problem with $k^2 = 2 \pi^2$ using Finite Element Method, with $f(x, y) = 10 e^{-100(x^2 + y^2)}$, and there's no error. The matrix is of full rank. So I don't understand why we still have a solution, which is maybe unique (the matrix is of full rank). Where did I make mistake?
Thank you.

Numerical eigenvalues are not the same as the exact eigenvalues (but close). When you refine the mesh the numerical eigenvalues approach the exact ones and the problem becomes ill-conditioned (but still of full rank).
