## 1. Introduction

[2] Significant research efforts have been expended on the development of ever more powerful electromagnetic solvers. Over the past two decades, the multilevel fast multipole algorithm (MLFMA) has proved to be a robust and error-controllable recipe to reduce the computational complexity of the matrix-vector product during the iterative Method of Moments (MoM) solution of boundary integral equations [*Chew et al.*, 2001]. Most efforts have been directed toward the solution of large three-dimensional perfect electric conducting (PEC) objects as, e.g., by *Velamparambil et al.* [2003] and *Ergül and Gürel* [2009].

[3] This paper is concerned with the development of an efficient MLFMA solver that can handle a 2D geometry illuminated by arbitrary plane waves in 3D space. This means that even though the geometry is infinitely long in, e.g., the *z* direction, field values still depend on the *z* coordinate in general. In the case of oblique plane wave incidence, this dependency is of the form *e*^{−jβz}, which leads to a coupling of the 2D transverse magnetic (TM) and transverse electric (TE) problem.

[4] The boundary integral equation that was employed and its classic MoM solution were first developed by *Olyslager et al.* [1993] to study the behavior of waveguides. In a later effort, *De Backer et al.* [1997] used the Impedance Matrix Localization (IML) method to sparsify the system matrix. This allowed for the application of the integral equation to the prediction of indoor wave propagation. Other applications can be found in research fields such as imaging and tomography [see, e.g., *Van den Bulcke and Franchois*, 2009; *Ngakosso et al.*, 1998]. In that context, the problem at hand is often referred to as ‘2.5-D’. This term however, is also frequently used for scattering problems in layered media, and is hence a possible source of confusion.

[5] In earlier work, we have proposed a kernel-independent, asynchronous, hierarchical parallel MLFMA applied to (pure) 2D TM scattering problems. The asynchronous algorithm allows for an efficient parallelization of simulations that involve multiple dielectric and/or PEC objects [*Fostier and Olyslager*, 2008b], while the hierarchical approach [*Ergül and Gürel*, 2008] allows for scalable parallel computations [*Fostier and Olyslager*, 2008a]. The term *kernel-independent* denotes that the parallel framework makes no assumptions about the electromagnetic MoM scheme that is used. In work by *Peeters et al.* [2008], the same framework was used for 3D broadband electromagnetic shielding problems. In this contribution, we report the application of this parallel framework to the coupled TE/TM integral equation.

[6] Compared to 3D simulations, 2D solvers allow for significantly larger simulations in terms of wavelengths. However, because the interactions between the discretized elements have a longer action range due to the slower decay of the 2D Green function, a high precision of the numerical methods is imperative. To the best of the authors knowledge, this paper is the first one to deal with the application of the MLFMA to coupled TE/TM simulations.

[7] This paper is organized as follows: in section 2, we outline the integral equation and its MoM solution. In section 3, the MLFMA is developed. Special attention is devoted to the memory-efficient storage of the lowest-level aggregation and disaggregation matrices. Also, a short outline of the parallel methodology and low-frequency stabilization is given. Finally, in section 4.1, we demonstrate the accuracy of the solver for different angles of oblique plane wave incidence and for very large scale examples with a diameter of 700,000*λ*.