On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients

We consider a scalar diffusion equation with a sign-changing coefficient in its principle part. The well-posedness of such problems has already been studied extensively provided that the contrast of the coefficient is non-critical. Furthermore, many different approaches have been proposed to construct stable discretizations thereof, because naive finite element discretizations are expected to be non-reliable in general. However, no explicit example proving the actual instability is known and numerical experiments often do not manifest instabilities in a conclusive manner. To this end we construct an explicit example with a broad family of meshes for which we prove that the corresponding naive finite element discretizations are unstable. On the other hand, we also provide a broad family of (non-symmetric) meshes for which we prove that the discretizations are stable. Together, these two findings explain the results observed in numerical experiments.