Singular value decompositions for single-curl operators in three-dimensional Maxwell's equations for complex media

Ruey Lin Chern, Han En Hsieh, Tsung Ming Huang, Wen Wei Lin, Weichung Wang

研究成果: 雜誌貢獻期刊論文同行評審

7 引文 斯高帕斯(Scopus)

摘要

This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest positive eigenvalues, and the main challenge is that the coefficient matrix in the discrete Maxwell equation is indefinite and degenerate. To overcome this difficulty, we derive a singular value decomposition (SVD) of the discrete single-curl operator and then explicitly express the basis of the invariant subspace corresponding to the nonzero eigenvalues of the GEP. Consequently, we reduce the GEP to a null space free standard eigenvalue problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP and can be solved by the shift-and-invert Arnoldi method without being disturbed by the null space. Furthermore, the basis of the eigendecomposition is chosen carefully so that we can apply fast Fourier transformation (FFT-) based matrix vector multiplication to solve the embedded linear systems efficiently by an iterative method. For chiral and pseudochiral complex media, which are of great interest in magnetoelectric applications, the NFSEP can be further transformed to a null space free GEP whose coefficient matrices are Hermitian and Hermitian positive definite (HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method without shifting. Furthermore, the embedded linear system can be solved efficiently by using the conjugate gradient method without preconditioning and the FFT-based matrix vector multiplications. Numerical results are presented to demonstrate the efficiency of the proposed methods.

原文英語
頁(從 - 到)203-224
頁數22
期刊SIAM Journal on Matrix Analysis and Applications
36
發行號1
DOIs
出版狀態已發佈 - 2015

ASJC Scopus subject areas

  • Analysis

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