In this work we present an adaptive boundary-integral equation method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. The indefiniteness of the permittivity tensor gives rise to preferential wave radiation within the propagating cone for the hyperbolic media, and this induces sharp transition for the solution of the integral equation across the cone boundary when waves start to decay or grow exponentially. In order to avoid a global refined mesh over the whole boundary, we employ a two-level a posteriori error estimator and an adaptive mesh refinement procedure to resolve the singularity locally for the solution of the integral equation. Such an adaptive procedure allows for the reduction of the number of the degrees of freedom significantly for the integral equation solver while achieving desired accuracy for the solution. In addition, to resolve the fast transition of the fundamental solution and its derivatives accurately across the propagation cone boundary, adaptive numerical quadrature rules are applied to evaluate the integrals for the stiffness matrices. Finally, to formulate the integral equations over the boundary we also derive the limits of layer potentials and their derivatives in the hyperbolic media when the target points approach the boundary.

在这项工作中，我们提出了一种自适应边界积分方程方法，用于计算双曲超材料中波相互作用的电磁响应。介电常数张量的不定性在双曲介质的传播锥内产生优先波辐射，当波开始衰减或呈指数增长时，这会导致积分方程在锥边界上的解的急剧转变。为了避免整个边界上的全局细化网格，我们采用两级后验误差估计器和自适应网格细化程序来解决积分方程解的局部奇异点。这种自适应过程允许显着减少积分方程求解器的自由度数量，同时实现解的所需精度。此外，为了准确解决基本解及其导数跨传播锥边界的快速过渡，应用自适应数值求积规则来评估刚度矩阵的积分。最后，为了制定边界上的积分方程，我们还推导出了当目标点接近边界时双曲介质中层电位及其导数的极限。应用自适应数值求积规则来评估刚度矩阵的积分。最后，为了制定边界上的积分方程，我们还推导出了当目标点接近边界时双曲介质中层电位及其导数的极限。应用自适应数值求积规则来评估刚度矩阵的积分。最后，为了制定边界上的积分方程，我们还推导出了当目标点接近边界时双曲介质中层电位及其导数的极限。