Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-25T13:48:04.870Z Has data issue: false hasContentIssue false
  • English
  • Français

An Iterative Two-Grid Method of A Finite Element PML Approximation for the Two Dimensional Maxwell Problem

Published online by Cambridge University Press:  03 June 2015

Chunmei Liu ,
Shi Shu ,
Yunqing Huang ,
Liuqiang Zhong  and
Junxian Wang
Chunmei Liu*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
Shi Shu*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
Liuqiang Zhong*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Junxian Wang*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
*
URL: http://math.xtu.edu.cn/myphp/math/personal/shushi/, Email: liuchunmei8080@yahoo.com.cn
Corresponding author. Email: shushi@xtu.edu.cn
Email: huangyq@xtu.edu.cn
Email: zhong@scnu.edu.cn
Email: xianxian.student@sina.com
Get access

Abstract

In this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer(PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of H(grad) variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete H(curl)-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.

Keywords

65F1065N3078A46Maxwell scatteringedge finite elementPMLiterative two-grid method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 2 , April 2012 , pp. 175 - 189
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bécache, E. AND Joly, P., On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 87–119.Google Scholar
[2] Beck, R., Deuflhard, P., Hiptmair, R., Hoppe, R. H. W. and Wohlmuth, B., Adaptive multilevel methods for edge element discretizations of Maxwell’s equations, Surveys. Math. Indust., 8 (1999), pp. 271–312.Google Scholar
[3] BéRenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Com-put. Phys., 114 (1994), pp. 185–200.Google Scholar
[4] BéRenger, J. P., Perfectly matched layer (PML)for computational electromagnetics, Volume 2 of Synthesis Lectures on Computational Electromagnetics, Morgan & Claypool, 2007.Google Scholar
[5] Bramble, J. H. and Pasciak, J. E., Analysis of a cartesian PML approximation to the three dimensional electromagnetiac wave scattering problem, Preprint, 2010.Google Scholar
[6] Botros, Y. and Volakis, J., Preconditioned generalized minimal residual iterative scheme for perfectly matched layer terminated applications, IEEE Microw. Guided. W., 9 (1999), pp. 45–47.Google Scholar
[7] Botros, Y. and Volakis, J., Perfectly matched layer termination for finite-element meshed: implementation and application, Microw. Opt. Tech. Lett., 23 (1999), pp. 166–172.Google Scholar
[8] Banks, H. and Browning, B., Time domain electromagnetic scattering using finite elements and perfectly matched layers, Comput. Methods. Appl. Mech. Eng., 194 (2005), pp. 149–168.CrossRefGoogle Scholar
[9] Bao, G. and Wu, H., Convergence analysis of the perfectly matched layer problems for time harmonic Maxwell’s euqations, SIAM J. Numer. Anal., 43 (2005), pp. 2121–2143.Google Scholar
[10] Chew, W. and Weedon, W., A 3d perfectly matched medium for modified Maxwell’s equations with streched coordinates, Microw. Opt. Tech. Lett., 13 (1994), pp. 599–604.Google Scholar
[11] Monk, P., Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2003.Google Scholar
[12] Hiptmair, R., Multigrid method For Maxwell’s equations, SIAM J. Numer. Anal., 36 (1998), pp.204–225.Google Scholar
[13] Jin, J., Shu, S. and Xu, J., A two-grid discretization method for decoupling systems of partial differntial equations, Math. Comput., 75 (2006), pp. 1617–1626.Google Scholar
[14] Hiptmair, R. and Xu, J., Nodal auxiliary spaces preconditions in H(curl) and H(div) spaces, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509.Google Scholar
[15] Huang, Y. and Chen, Y., A multilevel iterative method for finite element solutions to singular two-point boundary value problems, Natur. Sci. J. Xiangtan Univ., 16 (1994), pp. 23–26.Google Scholar
[16] Mu, M. and Xu, J., A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), pp. 1801–1813.Google Scholar
[17] Teixeira, F. L. and Chew, W. C., Differentialforms, metrics, and the reflectionless absorption of electromagnetic waves, J. Electromagn. Waves. Appl., 13 (1999), pp. 665–686.Google Scholar
[18] Xu, J., Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems, in Proceedings of the 5th International Symposium on Domain Decomposition Methods for Partial Differential Equations, Chan, T., Keyes, D., Meurant, G., Scroggs, J. and Voigt, R. (eds), Siam, Philadelphia, 1992, pp. 106–118.Google Scholar
[19] Xu, J., Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), pp. 1759–1777.CrossRefGoogle Scholar
[20] Xu, J. and Cai, X., A preconditioned gmres method for nonsymmetric or indefinite problems, Math. Comput., 59 (1992), pp. 311–319.Google Scholar
[21] Xu, J. and Zhou, A., Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math., 14 (2001), pp. 293–327.Google Scholar
[22] Xu, J. and Zhou, A., A two-grid discretization scheme for eigenvalue problems, Math. Com-put., 70 (2001), pp. 17–25.Google Scholar