Multiscale Methods for wave propagation in materials with sign-changing coefficients
For computational scientists modeling metamaterials, this provides a theoretically grounded multiscale method for a class of previously intractable PDEs with sign-changing coefficients.
The paper extends the CEM-GMsFEM to handle time-harmonic electromagnetic wave problems with sign-changing coefficients, establishing inf-sup stability and a priori error estimates, with numerical results showing effectiveness and robustness.
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.