On some 1D nonlocal models with coefficients changing sign

TL;DR

This paper investigates one-dimensional nonlocal elliptic transmission problems with sign-changing coefficients, establishing theoretical properties, proposing a numerical method, and demonstrating convergence and stability through simulations.

Contribution

It introduces a weak T-coercivity result for nonlocal problems with sign-changing coefficients and develops a finite element scheme with proven convergence.

Findings

Proved a weak T-coercivity result for the fractional problem.

Designed a finite element discretization that converges as s→1− and h→0.

Numerical simulations confirm stability and consistency of the proposed method.

Abstract

In this work, we study one-dimensional nonlocal elliptic transmission problems with piecewise constant coefficients that may change sign across an interface. In the local setting, we recall the T-coercive structure of the problem and characterize the critical contrast case. In the nonlocal setting, we focus on a simplified configuration in which the cross-interaction coefficient vanishes. Under this assumption, we prove a weak T-coercivity result for the global fractional problem and introduce a reconstructed formulation based on an explicit interface lifting. Then, we consider a simplified finite element discretization of the reconstructed model and prove its convergence toward the classical local transmission problem as the fractional parameter $s\to 1^-$ and the mesh size $h\to 0^+$. Numerical simulations in 1D illustrate the stability and consistency of the method, and a preliminary two-dimensional extension is presented as an exploratory perspective.