A new method called Expanded Cholesky Method (ECM) is proposed in this paper. The method can be used to decompose sparse symmetric non-positive-definite finite element (FEM) matrices. There are some advantages of the ECM, such as low storage, simplicity and easy parallelization. Based on the method, multifrontal (MF) algorithm is applied in non-positive-definite FEM computation. Numerical results show that the hybrid ECM/MF algorithm is stable and effective. In comparison with Generalized Minimal Residual Method (GMRES) in FEM electromagnetic computation, hybrid ECM/MF technology has distinct advantages in precision. The proposed method can be used to calculate a class of non-positive-definite electromagnetic problems.
2. Zhang, J. J., Y. Luo, S. Xi, H. Chen, L.-X. Ran, B.-I. Wu, and J. A. Kong, "Directive emission obtained by coordinate transformation," Progress In Electromagnetics Research, Vol. 81, 437-446, 2008.
3. Soleimani, M., C. N. Mitchell, R. Banasiak, R. Wajman, and A. Adler, "Four-dimensional electrical capacitance tomography imaging using experimental data," Progress In Electromagnetics Research, Vol. 90, 171-186, 2009.
4. Ozgun, O. and M. Kuzuoglu, "Finite element analysis of electromagnetic scattering problems via iterative leap-field domain decomposition method," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 2--3, 251-266, 2008.
5. Hu, J. X. and B. D. Ge, "Study on conformal FDTD for electromagnetic scattering by targets with thin coating," Progress In Electromagnetics Research, Vol. 79, 305-319, 2008.
6. Kalaee, P. and J. Rashed-Mohassel, "Investigation of dipole radiation pattern above a chiral media using 3D BI-FDTD approach," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 1, 75-86, 2009.
7. Liu, H. and H. W. Yang, "FDTD analysis of magnetized ferrite sphere," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 17--18, 2399-2406, 2008.
8. Swillam, M. A. and M. H. Bakr, "Full wave sensitivity analysis of guided wave structures using FDTD," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 16, 2135-2145, 2008.
9. Dehdasht-Heydari, R., H. R. Hassani, R. A. Mallahzadeh, and , "A new 2--18 GHz quad-ridged horn antenna," Progress In Electromagnetics Research, Vol. 81, 183-195, 2008.
10. Kasabegouda, V. G. and J. K. Vinoy, "Broadband suspended microstrip antenna for circular polarization," Progress In Electromagnetics Research, Vol. 90, 353-368, 2009.
11. Carpentieri, B., "Fast iterative solution methods in electromagnetic scattering," Progress In Electromagnetics Research, Vol. 79, 151-178, 2008.
12. Zhao, L., T. J. Cui, and W. D. Li, "An efficient algorithm for EM scattering by electrically large dielectric objects using MR-QEB iterative scheme and CG-FFT method," Progress In Electromagnetics Research, Vol. 67, 67-341, 2009.
13. Babolian, E., Z. Masouri, and S. Hatamzadeh-Varmazyar, "NEW direct method to solve nonlinear volterra-fredholm integral and integro-di®erential equations using operational matrix with block-pulse functions ," Progress In Electromagnetics Research B, Vol. 8, 59-76, 2008.
14. Schreiber, R., "A new implementation of sparse Gaussian elimination," ACM Transactions on Mathematical Software , Vol. 8, 256-276, 1982.
15. Duff, S. I. and K. J. Reid, "The multifrontal solution of indefinite sparse symmetric linear equations," ACM Transactions on Mathematical Software, Vol. 9, 302-325, 1983.
16. Chen, R. S., D. X. Wang, E. K. N. Yung, and J. M. Jin, "Application of the multifrontal method to the vector FEM for analysis of microwave filters," MOTL, Vol. 31, 465-470, 2001.
17. Hao, X. S., Theoretical Research and Implementation of Technology of Multifrontal Method, Solid Mechanics, Southeast University, China, 2003.
18. Liu, J., "The multifrontal method for sparse matrix solution: Theory and practice," SIAM Review, 82-109, 1992.
19. Zhuang, W., X. P. Feng, L. Mo, and R. S. Chen, "Multifrontal method preconditioned sparse-matrix/canonical grid algorithm for fast analysis of microstrip structure," APCP, 2005.
20. Wiping, Y., Numerical Analysis, Southeast University Press, 1992.
21. George, A. and J. W. Liu, Computer Solution of Large Sparse Positive Definite, Prentice Hall Professional Technical Reference, 1981.
22. Ashcraft, C. and J. Liu, "Generalized nested dissection: Some recent progress," Proceedings of Fifth SIAM Conference on Applied Linear Algebra, Snowbird, Utah, 1994.
23. Amestoy, P. R., A. T. Davis, and I. S. Duff, "An approximate minimum degree ordering algorithm," SIAM Journal on Matrix Analysis and Applications, Vol. 17, 886-905, 1996.
24. Liu, J., "The role of elimination trees in sparse factorization," SIAM Journal on Matrix Analysis and Applications, Vol. 11, 134, 1990.