## 1. Introduction

[2] The finite element (FE) method is a very powerful technique for the modal analysis of electromagnetic (EM) waveguides (WG) [*Rahman and Davies*, 1984; *Lee et al.*, 1991; *Dillon et al.*, 1993]. Since it is just the WG cross sections that need to be discretized, single solutions are computationally inexpensive. In many applications, however, the characteristics of WG modes are to be determined over wide frequency bands. Since modal field patterns may be frequency dependent and the corresponding propagation coefficients highly dispersive, and because dispersion curves may feature bifurcations, crossover points, and coupled-mode sections, the broadband analysis of EM waveguides typically requires a large number of FE computations, at different frequency points. In such cases, computer runtime is still a limiting factor. Methods of model order reduction (MOR) [*Polstyanko et al.*, 1997; *Bertazzi et al.*, 2002; *Lee et al.*, 2005; *Schultschik et al.*, 2008; *Ahmadloo and Dounavis*, 2008] provide a means to speed up the solution times of frequency sweeps very significantly, at little additional error. Most MOR methods are based on projection [*Antoulas*, 2005]. In the FE context, they can be viewed as restrictions of the trial and test spaces to low dimensional subspaces of global shape functions. Since the dimension of a reduced-order model (ROM) is typically in the tens, solution rates are very high: some hundreds to thousands per second, on a personal computer.

[3] Depending on the way the basis functions are constructed, MOR methods can be categorized as either single-point or multipoint approaches. Single-point methods [*Polstyanko et al.*, 1997; *Lee et al.*, 2005; *Ahmadloo and Dounavis*, 2008] match higher order derivatives of the considered eigenvalues and eigenvectors about a single expansion point. Hence, they need to factorize the FE matrix only once to set up the shift-and-invert preconditioner for the eigenvalue solver, which makes them computationally efficient. In case of very broad frequency ranges, however, single-point methods may suffer from numerical cancelation [*Polstyanko et al.*, 1997; *Lee et al.*, 2005; *Ahmadloo and Dounavis*, 2008]. Multipoint methods [*Bertazzi et al.*, 2002; *Schultschik et al.*, 2008], on the other hand, employ several expansion points. They are numerically robust and very well suited for broadband applications. Numerical tests for the driven case [*Schultschik et al.*, 2009] have shown that multipoint methods require fewer shape functions than single-point methods to meet a given error limit.

[4] This paper introduces a multipoint MOR method employing an adaptive point placement scheme to control the error. To the authors knowledge, this is the first fully automatic multipoint MOR method for the modal analysis of EM waveguides. It extends the adaptive framework of *Schultschik et al.* [2009] from the driven case to parameterized eigenvalue problems and improves on the FE formulation of *Schultschik et al.* [2008]. The proposed strategy is based on successively bisecting the interval of worst error measure. To guide the adaptive process, an incremental error indicator for the propagation coefficient is provided. The numerical results of section 6 show that the suggested approach achieves exponential convergence rates, in accordance with the best theoretical estimates available [*Binev et al.*, 2011; *Buffa et al.*, 2011].

[5] The paper is organized as follows. Section 2 gives the problem setting and defines commonly used abbreviations. The FE formulation used in this work is summarized in section 3. Section 4 derives the MOR method and comments on the special case of homogeneous material properties. The adaptive point placement strategy, including the error indicator, is presented in section 5. The numerical examples of section 6 demonstrate the accuracy and speed of the suggested approach.