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

Martin Halla; Florian Oberender

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arXiv · 2504.07712

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

Martin Halla·Florian Oberender

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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.

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Martin Halla, Florian Oberender. 2025-04-10. On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients. https://arxiv.org/abs/2504.07712

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