Volume 4, Issue 2
An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem

Adv. Appl. Math. Mech., 4 (2012), pp. 175-189.

Published online: 2012-04

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• 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 $\boldsymbol{H}(curl)$-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

Maxwell scattering, edge finite element, PML, iterative two-grid method.

65F10, 65N30, 78A46

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• TXT
@Article{AAMM-4-175, author = {}, title = {An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {2}, pages = {175--189}, 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 $\boldsymbol{H}(curl)$-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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m11166}, url = {http://global-sci.org/intro/article_detail/aamm/113.html} }
TY - JOUR T1 - An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 175 EP - 189 PY - 2012 DA - 2012/04 SN - 4 DO - http://doi.org/10.4208/aamm.10-m11166 UR - https://global-sci.org/intro/article_detail/aamm/113.html KW - Maxwell scattering, edge finite element, PML, iterative two-grid method. AB -

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 $\boldsymbol{H}(curl)$-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.

Chunmei Liu, Shi Shu, Yunqing Huang, Liuqiang Zhong & Junxian Wang. (1970). An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem. Advances in Applied Mathematics and Mechanics. 4 (2). 175-189. doi:10.4208/aamm.10-m11166
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