Computer Physics Communications 181 (2010) 271–276 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Application of the incomplete Cholesky factorization preconditioned Krylov subspace method to the vector finite element method for 3-D electromagnetic scattering problems Liang Li, Ting-Zhu Huang ∗ , Yan-Fei Jing, Yong Zhang School of Applied Mathematics/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, PR China article info abstract Article history: Received 22 September 2008 Received in revised form 14 April 2009 Accepted 26 September 2009 Available online 30 September 2009 Keywords: Electromagnetic scattering Krylov subspace method Complex symmetric matrix Finite element method Incomplete Cholesky factorization Precondition The incomplete Cholesky (IC) factorization preconditioning technique is applied to the Krylov subspace methods for solving large systems of linear equations resulted from the use of edge-based finite element method (FEM). The construction of the preconditioner is based on the fact that the coefficient matrix is represented in an upper triangular compressed sparse row (CSR) form. An efficient implementation of the IC factorization is described in detail for complex symmetric matrices. With some ordering schemes our IC algorithm can greatly reduce the memory requirement as well as the iteration numbers. Numerical tests on harmonic analysis for plane wave scattering from a metallic plate and a metallic sphere coated by a lossy dielectric layer show the efficiency of this method.  2009 Elsevier B.V. All rights reserved. 1. Introduction The edge element-based finite element method (FEM) [1] is widely used in high-frequency electromagnetic field simulations, such as waveguide discontinuities, antennas, and scattering [2,3]. It can combine any geometrical adaptability and material general- ity for modeling geometry and materials of any composition [4]. As to scattering problems, we primarily solve the following vector wave equation: ▽×  1 μ r ▽×E sc  − k 2 0 ε r E sc = −▽ ×  1 μ r ▽×E inc  + k 2 0 E inc (1) with some absorbing boundary conditions, where E sc is the scat- tering field, E inc is the incident field, and μ r and ε r are relative permeability and permittivity, respectively. Many research papers can be found to solve this problem [5–7] with FEM. The analytical domain of a typical scattering problem is theoret- ically an infinite one. Therefore, an absorbing boundary conditions should be applied to truncate the infinite open domain into a finite numerical domain. The purpose of an absorbing boundary condi- tion is to absorb the outgoing electromagnetic waves so that there * Corresponding author. E-mail addresses: plum_liliang@uestc.edu.cn (L. Li), tzhuang@uestc.edu.cn, tingzhuhuang@126.com (T.-Z. Huang), yanfeijing@uestc.edu.cn (Y.-F. Jing), mathzy@uestc.edu.cn (Y. Zhang). are no reflections back into the finite element analysis computa- tional domain. A perfectly matched layer (PML) [8] which is backed by a perfect electric conductor (PEC) boundary condition is used in our applications. It is an artificial anisotropic material that is trans- parent and heavily lossy to incoming electromagnetic waves. PML can reduce the size of the computational domain significantly with very small numerical reflections and is considered superior to con- ventional radiation absorbing boundary conditions. Discretization of (1) will result in a linear system Ax = b (2) with A = (a ij ) n×n ∈ C n×n sparse complex symmetric (usually in- definite), x, b ∈ C n . The sparsity of the coefficient matrix is a merit of FEM superior to integral-equation (IE) methods [9]. When ap- plied to 3-D scattering problems, the number of unknowns in- creases rapidly as the size of the problem increases. Therefore, O (n)-storage schemes [10] and effective solving techniques should be developed to tackle the 3-D problems. To solve (2) effectively, some Krylov subspace methods [11] proposed for general linear systems can be applied, such as GMRES [12], BiCGStab [13], and QMR [14]. Making use of symmetry of the systems, van der Vorst and Melissen [15] proposed the conjugate orthogonal conjugate gradient (COCG) method, which is regarded as an extension of the conjugate gradient (CG) method [16]. The COCG method searches the approximate solution through a Krylov subspace, but without a minimum residual property. Moreover, be- cause of the conjugate orthogonalization, the COCG method has 0010-4655/$ – see front matter  2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2009.09.019