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

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.

Figures (10)

• Figure 1: Illustration of the nested meshes $\mathscr{K}_h$ and $\mathscr{K}_H$. A fine element $\tau$, two coarse elements $K_{i'}$ and $K_{i"}$, accompanied by their corresponding oversampling regions $K_{i'}^2$ and $K_{i"}^2$, are colored differently.
• Figure 2: (a) The coefficient profile and the marked coarse element. (b)--(d) The plot of the first/second/third eigenfunction corresponding to the marked coarse element.
• Figure 3: The subplots are marked as (x-y), where x can take a, b, or c, corresponding to the results for the first, second, or third eigenfunction, respectively. If y is 1, 2, or 3, the subplot displays the multiscale basis with $m$ oversampling layers, $m$ equal to y. Alternatively, if y is 4, the subplot shows the relative differences (y-axis) in the energy and $L^2$ norm of the multiscale bases between $m=8$ and $m=1,\dots,7$ (x-axis).
• Figure 4: Numerical results for the flat interface model with $\gamma=1/2$, where the relative errors of the proposed method with different numbers of oversampling layers $m$ and the $Q_1$ FEM are calculated w.r.t. the coarse mesh size $H$. Subplots (a) and (b) correspond to $(\sigma^+_*, \sigma^-_*)=(1.01, 1)$, which the relative errors are measured in the energy and $L^2$ norm, respectively. Similarly, subplots (c) and (d) correspond to the setting $(\sigma^+_*, \sigma^-_*)=(1, 1.01)$, following the same manner.
• Figure 5: Numerical results for the flat interface model with $(\sigma^+_*, \sigma^-_*, \gamma)=(1, 1.01, 0.49)$. Subplots (a) and (b) show the relative errors of the proposed method with different numbers of oversampling layers $m$ and the $Q_1$ FEM w.r.t. the coarse mesh size $H$, but measured in different norms. Subplots (c) and (d) display the actual pointwise differences between the reference solution and numerical solutions obtained by the $Q_1$ FEM with $H=1/80$ and the proposed method with $(H, m)=(1/80, 3)$.

Theorems & Definitions (16)

• Lemma 5.1
• Lemma 5.2
• proof
• Remark
• Proposition 5.3
• Lemma 5.4: ref. Chung2018
• Lemma 5.5
• proof
• Theorem 5.6
• Lemma 5.7