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

TL;DR

This paper introduces an adaptive boundary element method tailored for hyperbolic metamaterials, effectively handling directional wave propagation and singularities with localized mesh refinement and specialized quadrature, improving computational efficiency and accuracy.

Contribution

The work develops a novel adaptive boundary element approach with a two-level error estimator and adaptive quadrature for hyperbolic metamaterials, addressing directional wave decay and exponential growth.

Findings

Significant reduction in degrees of freedom while maintaining accuracy.

Effective resolution of sharp transitions across the propagation cone.

Accurate evaluation of layer potential limits in hyperbolic media.

Abstract

In this work we present an adaptive boundary element method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. One unique feature of hyperbolic metamaterial is the strongly directional wave in its propagating cone, which induces sharp transition for the solution of the integral equation across the cone boundary when wave starts 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 degree 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 stiff matrices. Finally, in order 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.