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

¹ Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, Karlsruhe, Germany
² Institut für Numerische und Angewandte Mathematik, Georg-August Universit¨at Göttingen, Göttingen, Germany

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 11 April 2025
Accepted: 21 December 2025

Abstract

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.