Changqing Ye

Eric T. Chung, Patrick Ciarlet, Xingguang Jin et al.

From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of \texttt{T}-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.

Eric T. Chung, Patrick Ciarlet, Xingguang Jin et al.

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.

Yingjie Zhou, Xiang Zhong, Changqing Ye et al.

Modeling complex spatial networks with multiscale heterogeneity poses significant mathematical and computational challenges. Lacking explicit PDE discretizations and facing excessive degrees of freedom, conventional methods often become computationally prohibitive. To address these challenges, we propose an efficient multiscale modeling for highly heterogeneous spatial networks. We construct multiscale basis functions tailored to spatial network models with heterogeneous edge weights and node degrees. A key novelty is that the proposed method doesn't introduce geometric parameters (such as Dirichlet nodes, distances, or mesh sizes), thereby preserving its purely algebraic nature and ensuring broad applicability. By incorporating a subgraph-wise estimate, we define a Poincaré constant $C_{\mathrm{po}}$ that renders the method independent of the underlying graph geometry. Then through an appropriate choice of the number of graph oversampling layers, we establish an $O(C_{\mathrm{po}})$ convergence independent of the local heterogeneity contrast. Notably, our scheme operates entirely within an algebraic framework, eliminating the need for Dirichlet nodes and positive-definiteness on specific matrices arising in the model. This flexibility enables the simulation of a wider range of physical models and accommodates various boundary conditions. Rigorous theoretical analyses are provided under suitable assumptions, and extensive numerical experiments validate the effectiveness of the proposed approach.