Multiscale modeling for a class of high-contrast heterogeneous sign-changing problems

TL;DR

This paper introduces a specialized multiscale finite element method for solving complex sign-changing PDEs related to negative-index metamaterials, demonstrating robustness and stability.

Contribution

It develops a tailored CEM-GMsFEM approach for sign-changing problems, including stability analysis and error estimates under technical assumptions.

Findings

Effective handling of complex coefficient profiles

Robustness against high contrast ratios

Theoretical stability and error bounds established

Abstract

The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the \texttt{T}-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.