## 1. Introduction

[2] Due to the growing interest in technology based on electromagnetics, such as wireless communication and photonics, it is important to be able to solve Maxwell's equations as quickly and as precisely as possible. When considering piecewise homogeneous media and perfect or imperfect conductors, one of the most popular and efficient simulation methods is the use of boundary integral equations and to discretize them using the method of moments (MoM) [*Harrington*, 1968]. In the MoM, the boundaries of the objects are divided into segments and for each segment the fields are expressed as a linear combination of basis functions. In this paper the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation for the boundary integral equations is used, which yields accurate solutions, but is ill conditioned [*Kolundzija and Djordjevic*, 2002]. Applying the MoM leads to a set of linear equations for which the system matrix is a dense matrix. A direct solution of this set requires (*N*^{3}) operations, with *N* the number of unknowns, which becomes unfeasible for large *N*. By solving the set of equations using Krylov-based, iterative methods, the complexity can be reduced to (*PN*^{2}), as each of the *P* iterations requires the evaluation of matrix-vector products. If the problem is well conditioned then *P* is much smaller than *N*. A further reduction of the complexity can be achieved by applying the Multilevel Fast Multipole Algorithm (MLFMA) [*Chew et al.*, 2001]. The MLFMA reduces the complexity of the matrix-vector multiplication from (*N*^{2}) to (*N* log *N*), allowing to solve problems with a large number of unknowns.

[3] If the number of unknowns *N* is large, the computational requirements exceed the capabilities of a single processor and a parallel MLFMA has to be invoked. A partitioning scheme for a scalable parallel MLFMA has been presented by *Fostier and Olyslager* [2008] and *Ergül and Gürel* [2009a].

[4] All the previously mentioned methods and algorithms are implemented in Nero2d, which is open source and can be downloaded free of charge at http://www.openfmm.net.

[5] This paper focuses on the simulation of the Luneburg lens [*Luneburg*, 1944; *Kay*, 1959; *Bogaert et al.*, 2007; *Parfitt et al.*, 2000] involving many unknowns, but at the same time exhibiting a complex geometry. Such problems require an MLFMA approach that remains stable and accurate at low frequencies, but at the same time remains truly broadband. Indeed, at the considered frequencies, the size of the MLFMA boxes on some of the lower levels is small with respect to the wavelength, whereas, at the higher levels, box sizes are comparable to the wavelength.

[6] In section 2, a very short recapitulation of the MLFMA for 2-D is given, indicating that the recently developed normalized plane-wave method (NPWM) [*Bogaert et al.*, 2006] is a robust way to solve the so-called low-frequency breakdown of the classical MLFMA.

[7] Section 3 considers the 2-D Luneburg lens geometry. The permittivity of this lens varies continuously as a function of its radius and focuses an incident plane wave into a single point on its surface. For the 2-D case, the solution of the problem can be written down analytically. As our 2-D MLFMA method can only handle objects with a constant permittivity and permeability, the Luneburg lens is divided into shells with constant material parameters, approximating the continuous lens. This results in a geometry where objects are embedded into other objects. An analytical solution for this discretized version of the lens is also available. By comparing the available analytical solutions and the numerical results from our NPWM-MLFMA, the validity of our numerical technique can be put to the test. A very similar approach to simulate a 3-D Luneburg lens has been presented by *Carayol and Stève* [2010]. Many other approaches to model complex geometries exist, [e.g., *Jordan et al.*, 2009]. Finally, section 4 presents some conclusions.