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

TL;DR

The paper investigates diffusion problems with sign-changing coefficients and shows that naive finite element discretizations can be unstable on broad mesh families, depending on the coefficient contrast $\kappa$ and the mesh-size ratio $r=h_+/h_-$. Through explicit spectral analysis on unbounded and bounded domains, it derives stability criteria via the diagonal terms $d_m=\sigma_-\mathfrak{f}_{\kappa,r}(t)$ and their bounded-domain analogues, identifying parameter regimes where the discretization remains stable or develops a nontrivial kernel as the mesh is refined. The main contribution is the explicit demonstration of instability for certain meshes and the identification of large families of meshes that guarantee stability, thereby explaining the zigzag error curves seen in numerical experiments and guiding mesh design and stabilization strategies for sign-changing-coefficient PDEs. The results highlight the delicate interplay between coefficient contrast, mesh geometry, and domain truncation in the reliability of naive discretizations, with practical implications for simulations of metamaterials and cloaking where sign-changing coefficients arise.

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.

Figures (4)

• Figure 1: Relative errors for critical values of $h_-$ (solid lines) and for nearly critical values of $h_-$ (dashed lines).
• Figure 2: Computed solution for $h_-\approx0.26048$ in a neighborhood of the interface $\{0\}\times(0,\pi)$.
• Figure 3: Relative errors for a mesh satisfying \ref{['eq:AssRyL_vv']}.
• Figure 4: Computed solution for $h_+\approx0.26048$ on a flipped mesh in a neighborhood of the interface $\{0\}\times(0,\pi)$.

Theorems & Definitions (62)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof