Stability for non-positive definite differential equation

Stability for non-positive definite differential equation

I am dealing with some physics problems, whose math abstraction is sort of
eigenvalue prob(second order differential equation discretized with FEM),
as usual.
Now the problem is, this system has a bunch positive eigenvalues as well
as the other negative ones.
Could anybody help how to manage the numerical stability of this
systems?(in physics languages, how to avoid spurious solution in this
system) ? seems related to the ellipticity of the differential operator?
thanks!