IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 6, JUNE 2017 7206404 A Geometric Frequency-Domain Wave Propagation Formulation for Fast Convergence of Iterative Solvers Matteo Cicuttin 1 , Lorenzo Codecasa 2 , Ruben Specogna 3 , Francesco Trevisan 3 1 Université Paris-Est, Cermics (ENPC), F-77455 Marne-la-Vallée, France 2 Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, I-20133 Milano, Italy 3 Dipartimento Politecnico di Ingegneria ed Architettura, Università di Udine, 33100 Udine, Italy The frequency-domain wave propagation problem is notoriously difficult to solve through iterative methods because it leads to a symmetric but indefinite linear system. For this reason, direct methods are usually employed at the expense of great memory usage. Convergence of iterative methods, however, could be obtained by regularizing the wave equation. We introduce such regularization in discrete geometric approach framework on polyhedral grids. Moreover, we extend the regularization to the impedance boundary condition. Index Terms— Convergence, discrete geometric approach (DGA), wave propagation. I. I NTRODUCTION T HE time-harmonic wave propagation problem in the spa- tial domain  is described by the well-known equation ∇× ν r ∇× e - ω 2 μ 0 ǫ 0 ǫ r e =-i ωμ 0 j (1) where e and j are complex vector-valued functions of the position, ω is the angular frequency, and ν and ǫ are the symmetric positive definite material tensors. Moreover, taking the divergence of (1), it is possible to obtain the so-called continuity equation -i ωǫ 0 ∇· ǫ r e =∇· j (2) which relates the divergence of the electric displacement field d = ǫ 0 ǫ r e with the divergence of the current j . One of the possible settings to discretize (1) and (2) is the discrete geometric approach (DGA) numerical scheme [1], similar to the finite integration technique [2] but suitable also for polyhedral meshes [3]. In DGA, the domain  is discretized with two polyhedral grids G, ˜ G . Grid G is named primal grid, while grid ˜ G is named dual grid and is obtained by the standard barycentric subdivision of G. Discrete differen- tial operators are then introduced on both grids. In particular, the discrete gradient G, curl C, and divergence D are, respec- tively, the node-edge, edge-face, and face-volume incidence matrix on the primal grid. The dual discrete differential opera- tors are then obtained by transposing the primal ones, in partic- ular, ˜ G = D T , ˜ C = C T , and ˜ D =-G T . We finally introduce the discrete Hodge operators M ν r (primal face to dual edge), M ǫ r (primal edge to dual face), and N -1 μ r ǫ 2 r (dual volume to primal node), which account for material properties [1]. This setting allows to discretize (1) as ( ˜ CM ν r C - ω 2 μ 0 ǫ 0 M ǫ r )U =-i ωμ 0 ˜ I (3) Manuscript received November 19, 2016; revised February 28, 2017; accepted March 6, 2017. Date of publication March 8, 2017; date of current version May 26, 2017. Corresponding author: Matteo Cicuttin (e-mail: matteo.cicuttin@uniud.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2017.2679341 where U and ˜ I are the arrays of the degrees of freedom corresponding to e and j . The entries of U are the circulations of e on the edges e i of G U i =  e i e · d l (4) while the entries of ˜ I are the fluxes of j on the faces ˜ f i of ˜ G I i =  ˜ f i j · d S. (5) It is well-known, however, that (3) leads to a symmetric indefinite linear system on which iterative solvers fail to con- verge [4]. For this reason, direct solvers are usually employed but at a great expense of memory resources. This in turn makes the solution of large scale wave propagation problems quite a challenging topic, frequently requiring approaches tailored for the specific problem [5]. In this paper, we propose a regu- larized formulation of (3) that could enable the convergence of iterative solvers on this type of problem. The improvement of our contribution over the existing literature [2], [6], [7] is that support for polyhedral grids is introduced and impedance boundary conditions are correctly handled. This paper is organized as follows. In Section II, the formu- lation for problems without impedance boundary conditions is presented. In Section III, the formulation is extended to work with impedance boundary conditions. Then, some observations about preconditioning are made. Finally, the numerical results are presented. II. REGULARIZED FORMULATION By taking the discrete divergence of (3), the discrete continuity equation is obtained -i ωǫ 0 ˜ DM ǫ r U = ˜ D ˜ I. (6) By premultiplying both of its sides by -M ǫ r GN -1 μ r ǫ 2 r / i ωǫ 0 and by adding it to (3), the continuity equation can be exploited to enforce the divergence of the electric displacement field [6], [7]. 0018-9464 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.