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.

翻译：变号问题的数学表述涉及一个散度形式的线性二阶偏微分方程，其系数在不同子域中可取得正值与负值。此类问题的物理背景源于负折射率超材料，既可作为嵌入常见材料基体中的夹杂物，亦可反之。本文提出一种基于约束能量最小化广义多尺度有限元方法（CEM-GMsFEM）的数值方法，专门针对变号问题设计。原CEM-GMsFEM中辅助空间的构建经过特殊调整以适应变号场景。数值结果表明，所提方法在处理复杂系数分布时具有有效性，且对系数对比率具有鲁棒性。在若干技术性假设下，通过应用 \texttt{T}-强制理论，我们建立了该方法的inf-sup稳定性并给出了先验误差估计。