Bowei Wu

Jared Weed, Bowei Wu, Jingfang Huang et al.

In this paper, we present an accurate numerical method for the time-harmonic Maxwell's equations for bi-periodic multilayered media with quasi-periodic incident waves using the Method of Fundamental Solutions in conjunction with a periodization scheme. Following an approach used in acoustic scattering problems, the electric and magnetic fields in each layer are expressed as a sum of near and distant interactions. The near interaction comprises interactions between the unit cell and its nearest neighboring copies, while the distant interaction is approximated by proxy source points placed on spheres surrounding the unit cell. Imposing continuity of tangential components at the layer interface, quasi-periodicity conditions on the walls of the unit cell, and Rayleigh-Bloch expansion for the radiation condition yields a system of equations for the unknown coefficients, which can be solved by Schur complement and a backward-stable solver. The scheme is verified with known solutions and exhibits exponential convergence close to $10^{-14}$ for both single and multiple interfaces. An example with 39 interfaces is presented to demonstrate the solver's performance. The paper provides promising results for extending this method to a fast and accurate boundary integral equation solver for many cutting-edge applications involving a large number of layers in electromagnetics and optics.

Alex H. Barnett, Bowei Wu, Shravan K. Veerapaneni

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899-2921. We create a "globally compensated" trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using very small numbers of discretization nodes per curve. We provide documented codes for other researchers to use.