An adaptive boundary element method for the transmission problem with hyperbolic metamaterials

Highlights: • An integral equation formulation for wave interactions in hyperbolic metamaterials. • Adaptive algorithm for solving the boundary integral equations. • The adaptive procedure reduces the number of the degrees of freedom significantly. 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.